HSN.VM.B.4Common CoreMathNumber and QuantityGrades 9-12
HSN.VM.B.4: Adding and Subtracting Vectors
In plain English: HSN.VM.B.4 is an advanced (+) Common Core number and quantity standard, usually taught in Precalculus, that asks students to add and subtract vectors. Students add end-to-end, component-wise and by the parallelogram rule, find the magnitude and direction of a sum from magnitude and direction form, see that magnitudes do not simply add, and subtract by adding the opposite vector.
(+) Add and subtract vectors.
a.Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
b.Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
c.Understand vector subtraction v - w as v + (-w), where -w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
Common Core State Standards for Mathematics · Domain: Vector and Matrix Quantities (VM) · Cluster: Perform operations on vectors. Also written as HSN-VM.B.4 or N-VM.4 · Official standard
Students add and subtract vectors in three representations: arrows placed end-to-end, arrows that share a tail (the parallelogram rule), and components. They then find the magnitude and direction of a sum when the vectors are given in magnitude and direction form, which is how forces and velocities are usually described, and they see why the magnitude of a sum is typically smaller than the sum of the magnitudes.
The last part of the lesson treats subtraction v - w as v + (-w), shows it graphically by connecting the tips of w and v, and computes it component-wise. Notation used on this page: a bold letter such as v names a vector, ||v|| is its magnitude, ⟨3, 4⟩ is component form, and a direction is an angle measured counterclockwise from the positive x-axis (east in context problems).
Learning Objectives
By the end of this lesson, students will be able to:
Add two vectors end-to-end, by the parallelogram rule, and component-wise, and explain why the three methods agree
Explain with an example why the magnitude of a sum is typically not the sum of the magnitudes, and say when it is
Find the magnitude and direction of the sum of two vectors given in magnitude and direction form
Describe the opposite vector -w and subtract vectors as v + (-w), graphically by connecting tips and component-wise
Prior Knowledge Required
Students should already be comfortable with:
Vectors as directed line segments with magnitude and direction, and the notation for vectors and magnitudes HSN.VM.A.1
Finding components by subtracting initial point from terminal point HSN.VM.A.2
Right-triangle trigonometry: sine, cosine and tangent of an angle HSG.SRT.C.8
The Pythagorean Theorem and the distance formula 8.G.B.8
Ask students to picture a walk on a city grid and answer without a calculator first, then check:
Warm-Up Prompt
"You walk 300 m east and then 400 m north. How far did you walk? How far are you from where you started? Would the second answer change if you walked north first?"
Students should find 700 m walked but only 500 m (a 3-4-5 triangle) from the start, in either order. Record the two walks as arrows placed tip-to-tail and name the straight arrow from start to finish the resultant. Keep both numbers on the board: the gap between 700 and 500 is the big idea of standard a.
Direct Instruction25 minutes
Part 1: Three ways to add (standard a). Draw u = ⟨4, 1⟩ and w = ⟨1, 3⟩ on a grid (Diagram 1). End-to-end: start w at the tip of u; the sum runs from the tail of u to the tip of w. Parallelogram rule: draw both from the same point, complete the parallelogram with dashed copies, and the sum is the diagonal from the shared tail. Component-wise: add the x-components and add the y-components. All three give ⟨5, 4⟩, and the parallelogram shows that u + w = w + u.
Write each vector in components: a vector with magnitude r and direction θ is ⟨r cos θ, r sin θ⟩.
Add the components: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩.
Find the magnitude of the sum: ||⟨x, y⟩|| = √(x² + y²).
Find the direction: tan θ = y/x. Use the signs of x and y to place θ in the right quadrant (add 180° when x is negative).
Part 2: Magnitudes do not simply add (standard a). Use the examples below. Compare ||sum|| with the sum of the magnitudes each time. They are equal only when the two vectors point the same way; the more the directions differ, the smaller the resultant, down to 0 for opposite vectors of equal length. This is the triangle inequality: one side of a triangle is shorter than the other two together.
Part 3: Magnitude and direction form (standard b). Work the force and plane examples with the four steps above. Point out that the angle between the two vectors, not their individual angles, decides how large the resultant is.
End-to-end and component-wise addition
Add u = ⟨4, 1⟩ and w = ⟨1, 3⟩. Draw w starting at the tip of u (Diagram 1), then add the matching components.
Equation: u + w = ⟨4 + 1, 1 + 3⟩ = ⟨5, 4⟩
Magnitude of a sum
a = ⟨6, 0⟩ has magnitude 6 and b = ⟨0, 8⟩ has magnitude 8. Is the magnitude of their sum 14?
Equation: a + b = ⟨6, 8⟩ and ||⟨6, 8⟩|| = √(36 + 64) = 10, not 14
Magnitude and direction form
Two ropes pull a crate across a floor: 12 N at 20° and 8 N at 75°. Find the magnitude and direction of the total force.
Equation: ⟨12 cos 20°, 12 sin 20°⟩ + ⟨8 cos 75°, 8 sin 75°⟩ ≈ ⟨11.28, 4.10⟩ + ⟨2.07, 7.73⟩ = ⟨13.35, 11.83⟩, about 17.84 N at 41.6°
Part 4: Subtraction (standard c). Define -w as the vector with the same magnitude as w pointing the opposite way, so w + (-w) = ⟨0, 0⟩, the zero vector. Then v - w means v + (-w). Show both pictures in Diagram 2: adding -w end-to-end, and the shortcut of drawing v and w from one point and connecting the tip of w to the tip of v. Check the order by asking: "What do I add to w to get v?"
Subtraction as adding the opposite
Find v - w for v = ⟨5, 2⟩ and w = ⟨2, 4⟩ in two ways (Diagram 2).
Equation: -w = ⟨-2, -4⟩, so v + (-w) = ⟨3, -2⟩, and component-wise ⟨5 - 2, 2 - 4⟩ = ⟨3, -2⟩
Resultant velocity in context
A small plane flies with airspeed 180 km/h heading north (90°). A wind blows toward the east (0°) at 40 km/h. Find the plane's speed and direction over the ground.
Pairs work four problems on grid paper, drawing each one before computing. Stop after each problem to compare pictures.
Guided practice problems and answers
Problem
Answer
Add ⟨-3, 5⟩ + ⟨7, -2⟩ end-to-end and component-wise. Compare the magnitudes.
⟨4, 3⟩, magnitude 5, while √34 + √53 ≈ 13.11
Draw ⟨2, 5⟩ and ⟨4, -1⟩ from the origin and sketch the parallelogram. Name both diagonals as vectors.
Sum ⟨6, 4⟩ from the origin; the other diagonal is a difference, ⟨2, 5⟩ - ⟨4, -1⟩ = ⟨-2, 6⟩
Find the resultant of 20 N at 0° and 20 N at 60°.
⟨30, 17.32⟩, about 34.64 N at 30°
Subtract ⟨6, -1⟩ - ⟨2, 3⟩ component-wise and by connecting tips.
⟨4, -4⟩
Listen for three errors: starting the second arrow at the tail instead of the tip, adding the magnitudes 20 + 20 = 40 N, and drawing the difference from the tip of the first vector to the tip of the second (which gives the opposite vector).
Independent Practice15 minutes
Students work alone on five problems and sketch each one:
⟨8, -3⟩ + ⟨-5, 7⟩ (answer ⟨3, 4⟩)
⟨-2, -6⟩ - ⟨-7, 1⟩ (answer ⟨5, -7⟩)
A walker goes 15 m east, then 9 m north. Find the displacement (about 17.49 m at 31.0°)
Add 10 units at 45° and 10 units at 135° (⟨0, 14.14⟩, about 14.14 units at 90°)
Write the opposite of ⟨3, -4⟩ and compare the magnitudes (⟨-3, 4⟩, both have magnitude 5)
For each magnitude and direction answer, students write one sentence comparing ||sum|| with the sum of the magnitudes.
Closure5-10 minutes
Exit ticket: (1) For ⟨1, 7⟩ and ⟨4, -2⟩, find the sum and the difference ⟨1, 7⟩ - ⟨4, -2⟩. (Answers: ⟨5, 5⟩ and ⟨-3, 9⟩.) (2) Two vectors have magnitudes 6 and 9. What are the largest and smallest possible magnitudes of their sum, and when does each happen? (15 when they point the same way, 3 when they point opposite ways.)
Differentiation Strategies
For Struggling Students
Give a component table with columns x and y and a row for each vector, so adding is adding down a column
Use two colors: the first vector in blue, the second in black, and the resultant always in green from the first tail to the last tip
Start magnitude and direction problems with perpendicular vectors so the resultant is the hypotenuse of a right triangle
For Advanced Students
Derive ||u + w||² = ||u||² + ||w||² + 2||u|| ||w|| cos θ, where θ is the angle between u and w, using components, and use it to check a resultant
Show that the two diagonals of the parallelogram spanned by u and w are u + w and u - w
Find three vectors of magnitude 5 whose sum is the zero vector, and describe the triangle they form end-to-end
Assessment Guidance
What to Look For
Check that every sum or difference comes with a sketch in which arrows meet tip to tail, or tail to tail for the parallelogram, and that the sketch agrees with the components. For magnitude and direction answers, look for a direction in the correct quadrant: a sum with a negative x-component cannot have a direction between -90° and 90°. For subtraction, ask students to state which tip the difference starts at, and why.
02
Classroom Activities
3 Activities
1
Walk the Vectors
20 minPairs
Pairs walk displacement vectors on a floor grid marked with masking tape (1 step = 1 floor tile or 30 cm), placing the second vector at the end of the first. They compare the distance walked with the straight-line distance, which makes standard a physical.
Vector Cards (6)
⟨3, 0⟩
⟨0, 2⟩
⟨-1, 3⟩
⟨2, -2⟩
⟨4, 1⟩
⟨-3, -1⟩
Procedure
Each pair draws two cards. Partner A walks the first vector, then the second, starting where the first ended; Partner B marks the start and finish with tape
Record the sum component-wise and measure the start-to-finish distance with a tape measure
Walk the same two cards in the other order and check that the finish point is the same
Repeat with two new cards. For example, ⟨3, 0⟩ then ⟨0, 2⟩ ends at ⟨3, 2⟩, about 3.6 tiles away after walking 5
Discussion Questions
Which pair of cards gave the biggest gap between distance walked and distance from the start? Which gave the smallest?
Can the straight-line distance ever be longer than the distance walked?
Which two cards add to a vector that points straight back toward the start of the first card?
Modification for Distance Learning
Students move a token on a shared digital grid, draw each card as an arrow, and measure the resultant with the grid's coordinates instead of a tape measure.
2
Paper Force Table
25 minGroups of 3
Each group gets 4 force cards, each showing two forces in magnitude and direction form. One student finds the resultant by a scale drawing with ruler and protractor (1 cm = 10 N), one by components, and one checks that the two answers agree within the accuracy of the drawing. This practices standard b.
Force Cards (4)
Card A: 50 N at 0° and 30 N at 60° (resultant 70 N at about 21.8°)
Card B: 40 N at 30° and 40 N at 150° (resultant 40 N at 90°)
Card C: 60 N at 0° and 25 N at 90° (resultant 65 N at about 22.6°)
Card D: 35 N at 45° and 35 N at 225° (resultant 0 N)
Procedure
Rotate roles after each card so that every student draws, computes and checks at least once
For each card, write the sum of the two magnitudes next to the resultant's magnitude
Groups post Card D on the board and explain the result to the class
Discussion Questions
On Card B the two forces are equal. Why does the resultant point straight up?
What is the angle between the two forces on each card, and how does it relate to how much smaller the resultant is than the sum of magnitudes?
How close did the scale drawing get to the component answer?
Challenge Variation
Give the resultant and one force (for example, resultant 70 N at 21.8° and 50 N at 0°) and ask groups to find the second force using subtraction.
3
Tip-to-Tip Subtraction
20 minPairs
Pairs subtract three pairs of vectors in three ways: component-wise, by adding the opposite vector end-to-end, and by connecting the tips. The goal is to settle the order question for standard c: the difference v - w starts at the tip of w.
Vector Pairs
v = ⟨6, 1⟩, w = ⟨2, 3⟩ (v - w = ⟨4, -2⟩)
v = ⟨-1, 4⟩, w = ⟨3, 2⟩ (v - w = ⟨-4, 2⟩)
v = ⟨0, -3⟩, w = ⟨-4, 1⟩ (v - w = ⟨4, -4⟩)
Procedure
Partner A computes v - w component-wise; Partner B draws v and w from the origin and connects the tips
Together, draw -w (same length, opposite direction) at the tip of v and confirm that v + (-w) ends at the same point as the component answer
Finally, compute w - v and compare it with v - w
Discussion Questions
How are v - w and w - v related, as arrows and as components?
Why does the tip-to-tip arrow you drew represent v - w and not w - v? Use the equation w + (v - w) = v in your answer.
What is v - v, and how would you draw it?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Two Ways to Add the Same Vectors
Both methods give the same sum u + w = ⟨4, 1⟩ + ⟨1, 3⟩ = ⟨5, 4⟩ (green marks the result). On the left, w starts at the tip of u. On the right, u and w share a tail and the sum is the diagonal of the parallelogram they span. The dashed sides are copies of u and w, which is why both pictures end at (5, 4). The grid is to scale: the green arrow is shorter than the two other arrows together.
Diagram 2: Subtracting Vectors Two Ways
Left: with v and w drawn from the same point, v - w runs from the tip of w to the tip of v, because w + (v - w) = v. Right: the same result comes from adding the opposite vector -w = ⟨-2, -4⟩ end-to-end to v. The green arrows are the same vector ⟨3, -2⟩ drawn in two positions.
04
Homework Assignment
~30 min
HSN.VM.B.4 Homework: Adding and Subtracting Vectors
Directions: Show all work and draw a sketch for every problem. Give magnitudes to two decimal places and directions to the nearest tenth of a degree, measured counterclockwise from the positive x-axis.
Part 1: Adding Vectors (Problems 1-2)
Let p = ⟨2, -5⟩ and q = ⟨-6, 2⟩. (a) Find p + q component-wise. (b) On grid paper, show the sum end-to-end and by the parallelogram rule. (c) Find ||p||, ||q|| and ||p + q||, and compare ||p + q|| with ||p|| + ||q||.
For each pair, find the sum and decide whether ||a + b|| = ||a|| + ||b||: (a) a = ⟨3, 4⟩, b = ⟨9, 12⟩ (b) a = ⟨3, 4⟩, b = ⟨4, 3⟩ (c) a = ⟨3, 4⟩, b = ⟨-6, -8⟩. Then explain in one or two sentences when the magnitude of a sum equals the sum of the magnitudes.
Part 2: Magnitude and Direction (Problems 3-4)
Two tugboats pull a barge. One pulls with 60 kN at 20°, the other with 45 kN at -30°. Find the magnitude and direction of the combined force. Is the combined force closer in direction to the stronger or the weaker tugboat?
A hiker walks 4.0 km at 60°, then 2.5 km at 160°. Find the magnitude and direction of the hiker's total displacement. Explain how you placed the direction in the correct quadrant.
Part 3: Subtracting Vectors (Problems 5-6)
Let v = ⟨7, -2⟩ and w = ⟨3, 4⟩. (a) Write -w and state its magnitude and how its direction compares with that of w. (b) Find v - w component-wise. (c) Show that v + (-w) gives the same vector. (d) Draw v and w from the origin, draw v - w by connecting the tips, and say which tip it starts at.
Two drones fly with velocities a = ⟨12, 5⟩ m/s and b = ⟨4, 11⟩ m/s. The velocity of drone A as seen from drone B is a - b. (a) Find a - b and its magnitude. (b) Find b - a and explain how the two differences are related. (c) Draw a and b from one point and show both differences by connecting tips.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Component Work
Sums and differences correct, with signs
One sign or arithmetic error
Components added incorrectly or missing
Sketches
End-to-end, parallelogram and tip-to-tip sketches drawn correctly and labeled
Sketches present but one is placed incorrectly
No sketches
Magnitude and Direction
Magnitudes and directions correct, direction in the correct quadrant
Magnitude correct but direction in the wrong quadrant
Magnitudes of the parts added
Explanation
Clear explanation of when magnitudes add and of the order in subtraction
Explanation partly correct
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Directions are measured counterclockwise from the positive x-axis. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Find ⟨3, -2⟩ + ⟨-5, 6⟩.
Answer: B
Add matching components: 3 + (-5) = -2 and -2 + 6 = 4, so the sum is ⟨-2, 4⟩. Choice A subtracts the second vector instead of adding it. Choice C makes a sign error in the y-component, and choice D multiplies the components.
Question 2 of 20 · Multiple Choice
To add u + w end-to-end, where do you place w?
Answer: C
End-to-end (tip-to-tail) addition starts w where u ends, and the sum runs from the tail of u to the tip of w. Choice A is the starting setup of the parallelogram rule, not end-to-end addition. Choice B connects tips, which is how a difference is drawn.
Question 3 of 20 · Multiple Choice
u and w are drawn from the same point and a parallelogram is completed. Which part of the parallelogram is u + w?
Answer: A
In the parallelogram rule the sum is the diagonal from the shared tail to the opposite corner, because going along u and then along the copy of w reaches that corner. Choice B is the other diagonal, which represents a difference such as u - w or w - u, not the sum.
Question 4 of 20 · Multiple Choice
Find the magnitude of ⟨9, 0⟩ + ⟨0, 12⟩.
Answer: D
The sum is ⟨9, 12⟩, and √(81 + 144) = √225 = 15. Choice A adds the magnitudes 9 + 12, which is only correct for vectors pointing the same way. Choice B subtracts them.
Question 5 of 20 · Multiple Choice
Which statement is true for every pair of vectors u and w?
Answer: B
The sum is the third side of a triangle whose other sides have lengths ||u|| and ||w||, so it can be no longer than those two together. It reaches that length only when the arrows line up in the same direction. Choice A treats magnitudes like numbers that simply add. Choice D fails when w points the same way as u.
Question 6 of 20 · Multiple Choice
Two forces act on an object: 7 N at 0° and 24 N at 90°. What is the magnitude of the total force?
Answer: C
The components add to ⟨7, 24⟩, and √(49 + 576) = √625 = 25 N. Choice A adds the magnitudes, and choice B subtracts them.
Question 7 of 20 · Multiple Choice
For the forces in the previous question (7 N at 0° and 24 N at 90°), what is the direction of the total force, to the nearest tenth of a degree?
Answer: A
The total force is ⟨7, 24⟩, so tan θ = 24/7 and θ ≈ 73.7°, in the first quadrant. Choice B uses 7/24, which is the angle measured from the y-axis. Choice C averages the two directions, which works only for equal magnitudes.
Question 8 of 20 · Multiple Choice
If w = ⟨-4, 9⟩, what is -w?
Answer: D
The opposite vector negates both components, so -w = ⟨4, -9⟩, and w + (-w) = ⟨0, 0⟩. Choices A and B negate only one component, so they point in a different direction, not the opposite one. Choice C swaps the components.
Question 9 of 20 · Multiple Choice
Find ⟨2, 7⟩ - ⟨5, -1⟩.
Answer: B
Subtract matching components: 2 - 5 = -3 and 7 - (-1) = 8. Choice C forgets that subtracting -1 adds 1. Choice D is ⟨5, -1⟩ - ⟨2, 7⟩, the difference in the wrong order. Choice A adds the vectors.
Question 10 of 20 · Multiple Choice
v and w are drawn from the same point. Which arrow represents v - w?
Answer: A
v - w is the vector you add to w to reach v: w + (v - w) = v. So it starts at the tip of w and ends at the tip of v. Choice B reverses the order and gives w - v. Choice C is the sum v + w.
Question 11 of 20 · Multiple Choice
A vector w has magnitude 8 and direction 50°. Which describes -w?
Answer: C
-w has the same magnitude as w and points in the opposite direction, which is 50° + 180° = 230°. Choice A is wrong because a magnitude is never negative. Choice B rotates by 90° instead of 180°, and choice D reflects the vector across the x-axis.
Question 12 of 20 · Multiple Choice
Two students pull a sled with forces of 150 N at 30° and 200 N at 90°. What are the magnitude and direction of the total force, to the nearest whole number?
Answer: D
Components: ⟨150 cos 30°, 150 sin 30°⟩ + ⟨0, 200⟩ ≈ ⟨129.9, 75⟩ + ⟨0, 200⟩ = ⟨129.9, 275⟩. The magnitude is √(129.9² + 275²) ≈ 304 N and tan θ = 275/129.9 gives θ ≈ 65°. Choice A adds the magnitudes and averages the angles. Choice B has the right magnitude but averages the angles, which is wrong because the forces are not equal. Choice C treats the forces as perpendicular (√(150² + 200²) = 250), but they are only 60° apart.
Question 13 of 20 · Multiple Choice
The vectors ⟨1, 4⟩ and ⟨5, 2⟩ are drawn from the origin, and a parallelogram is completed on them. Where is the fourth vertex?
Answer: B
The fourth vertex is the tip of the diagonal from the origin, which is the sum ⟨1, 4⟩ + ⟨5, 2⟩ = ⟨6, 6⟩. Choice A is the difference ⟨5, 2⟩ - ⟨1, 4⟩. Choice D is half of the sum, the point where the diagonals cross.
Question 14 of 20 · Multiple Choice
For v = ⟨6, 2⟩ and w = ⟨1, -10⟩, what is ||v - w||?
Answer: A
v - w = ⟨6 - 1, 2 - (-10)⟩ = ⟨5, 12⟩, and √(25 + 144) = 13. Choice B comes from ⟨5, -8⟩, which adds the y-components instead of subtracting them. Choice C is ||v + w|| = ||⟨7, -8⟩||, the magnitude of the sum. Choice D uses only the x-component.
Question 15 of 20 · Short Answer
Let u = ⟨-3, 4⟩ and w = ⟨8, 2⟩. Find u + w component-wise and describe how to draw it end-to-end starting at the origin.
u + w = ⟨-3 + 8, 4 + 2⟩ = ⟨5, 6⟩. Draw u from (0, 0) to (-3, 4), then draw w starting at (-3, 4): 8 right and 2 up, ending at (5, 6). The sum is the arrow from (0, 0) to (5, 6).
Question 16 of 20 · Short Answer
Give an example of two vectors u and w with ||u|| = 5 and ||w|| = 5 but ||u + w|| ≠ 10, and explain why this happens.
One example: u = ⟨5, 0⟩ and w = ⟨0, 5⟩ give u + w = ⟨5, 5⟩ with magnitude √50 ≈ 7.07. Placed end-to-end, the two vectors and their sum form a triangle, and one side of a triangle is shorter than the other two together. The magnitudes add to 10 only when the vectors point the same way.
Question 17 of 20 · Short Answer
u has magnitude 30 and direction 0°. w has magnitude 30 and direction 120°. Find the magnitude and direction of u + w.
u = ⟨30, 0⟩ and w = ⟨30 cos 120°, 30 sin 120°⟩ = ⟨-15, 15√3⟩ ≈ ⟨-15, 25.98⟩. The sum is ⟨15, 25.98⟩, with magnitude √(225 + 675) = √900 = 30 and direction tan⁻¹(25.98/15) = 60°. The two vectors and their sum form an equilateral triangle.
Question 18 of 20 · Short Answer
Let v = ⟨1, 5⟩ and w = ⟨4, -1⟩, both drawn from the origin. Describe the arrow that represents v - w: where it starts, where it ends, and its components.
It starts at the tip of w, (4, -1), and ends at the tip of v, (1, 5). Its components are ⟨1 - 4, 5 - (-1)⟩ = ⟨-3, 6⟩, so its magnitude is √45 = 3√5 ≈ 6.71.
Question 19 of 20 · Short Answer
Let v = ⟨-6, 3⟩ and w = ⟨-2, -5⟩. Write -w, then show that v + (-w) equals v - w computed component-wise.
-w = ⟨2, 5⟩, with the same magnitude √29 as w and the opposite direction. v + (-w) = ⟨-6 + 2, 3 + 5⟩ = ⟨-4, 8⟩. Component-wise, v - w = ⟨-6 - (-2), 3 - (-5)⟩ = ⟨-4, 8⟩, the same vector.
Question 20 of 20 · Short Answer
Forces of 9 N at 10° and 12 N at 100° act on the same point. Use the parallelogram rule to explain why the total force has magnitude 15 N, then find its direction.
The directions differ by 90°, so the parallelogram on the two forces is a rectangle, and the resultant is its diagonal: √(9² + 12²) = 15 N. The diagonal makes an angle tan⁻¹(12/9) ≈ 53.1° with the 9 N force, so its direction is 10° + 53.1° ≈ 63.1°. Components confirm it: ⟨9 cos 10° + 12 cos 100°, 9 sin 10° + 12 sin 100°⟩ ≈ ⟨6.78, 13.38⟩.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSN.VM.B.4 mean?
HSN.VM.B.4 asks students to add and subtract vectors. In detail, students add vectors end-to-end, component-wise and by the parallelogram rule (4a), find the magnitude and direction of a sum of vectors given in magnitude and direction form (4b), and subtract by adding the opposite vector, both on a graph and component-wise (4c).
Is HSN.VM.B.4 taught in Precalculus or Algebra 2?
It is usually taught in Precalculus, often in a vectors unit next to trigonometry. The (+) marks it as an advanced standard, additional mathematics for students who take advanced courses. Physics courses use the same skills for forces and velocities.
Why is the magnitude of a sum of vectors not the sum of the magnitudes?
Because the vectors usually point in different directions. Placed end-to-end, two vectors and their sum form a triangle, and one side of a triangle is shorter than the other two combined. For example, walking 300 m east and then 400 m north leaves you 500 m from the start, not 700 m.
When does the magnitude of a sum equal the sum of the magnitudes?
Only when the two vectors point in the same direction (or one of them is the zero vector). If they point in opposite directions, the magnitude of the sum is the difference of the magnitudes. Every other angle between them gives something in between.
Are end-to-end addition and the parallelogram rule the same thing?
Yes, they give the same vector. The parallelogram rule places both vectors at one point; its far side is a copy of the second vector starting at the tip of the first, which is exactly the end-to-end picture. The parallelogram also shows that u + w = w + u.
How do you find the direction of the sum of two vectors?
Write the sum in components ⟨x, y⟩ and use tan θ = y/x. A calculator's inverse tangent only returns angles between -90° and 90°, so when x is negative, add 180°. A quick sketch of the sum shows which quadrant the answer belongs in.
How do I know which way to draw v - w?
Draw v and w from the same point. The difference v - w goes from the tip of w to the tip of v, because it is the vector you add to w to get v. Drawing it the other way gives w - v, the opposite vector.
What are common mistakes in adding and subtracting vectors?
Common ones are adding magnitudes instead of components, starting the second arrow at the tail of the first instead of its tip, putting the direction of a sum in the wrong quadrant, dropping a sign when subtracting a negative component, and reversing the order in a difference.
Do students need the Law of Cosines for HSN.VM.B.4?
No. Components and right-triangle trigonometry are enough to find the magnitude and direction of any sum. The Law of Cosines (HSG.SRT.D.11) gives a second method from the triangle formed end-to-end, and it is a good check for students who already know it.
Where do students use vector addition and subtraction later?
In physics for net force, displacement and relative velocity, in Calculus and beyond for motion in the plane, and in computer graphics and navigation. Within Common Core, it leads to solving velocity problems with vectors (HSN.VM.A.3) and to matrices acting on vectors (HSN.VM.C.11).
07
Related Standards
6 standards
These standards connect to HSN.VM.B.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSN.VM.A.1Prerequisite
Recognize vectors as having magnitude and direction and use vector notation
