HSN.VM.A.2Common CoreMathNumber and QuantityGrades 9-12
HSN.VM.A.2: Finding Vector Components from Initial and Terminal Points
In plain English: HSN.VM.A.2 is an advanced (+) Common Core number and quantity standard that asks students to find the components of a vector by subtracting the coordinates of its initial point from the coordinates of its terminal point. A vector from (x1, y1) to (x2, y2) has components ⟨x2 - x1, y2 - y1⟩, the horizontal and vertical change. It is usually taught in Precalculus.
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
Common Core State Standards for Mathematics · Domain: Vector and Matrix Quantities (VM) · Cluster: Represent and model with vector quantities. Also written as HSN-VM.A.2 or N-VM.2 · Official standard
Students learn to write a vector in component form by subtracting coordinates: for a vector with initial point (x1, y1) and terminal point (x2, y2), the components are ⟨x2 - x1, y2 - y1⟩. The x-component is the horizontal change and the y-component is the vertical change as you travel from the tail of the arrow to its head. Students see that the order of subtraction sets the direction, that equal vectors in different places have the same components, and that the same rule works in three dimensions.
How vectors are written here. Bold letters such as v are vectors, and AB in bold is the vector that starts at A and ends at B. Components go inside angle brackets, as in ⟨4, 5⟩, so that a vector is not confused with the point (4, 5). On paper, students mark a vector with an arrow above the letter or letters.
Learning Objectives
By the end of this lesson, students will be able to:
Find the components of a vector by subtracting the coordinates of its initial point from the coordinates of its terminal point
Interpret the sign and size of each component as a horizontal or vertical change
Explain why reversing the initial and terminal points gives the opposite vector, and why equal vectors have equal components
Work backward from components and one endpoint to the other endpoint, in two and three dimensions
Prior Knowledge Required
Students should already be comfortable with:
Plotting points in all four quadrants and finding horizontal and vertical distances 6.NS.C.8
Subtracting signed numbers, including subtracting a negative 7.NS.A.1
Vectors as directed line segments with an initial and a terminal point HSN.VM.A.1
Draw a coordinate grid on the board and label it as the floor of a warehouse, 1 unit per meter. Pose the question below and have students answer on mini whiteboards.
Warm-Up Prompt
"A delivery robot rolls from (2, 3) to (7, 1). How many meters did it move left or right, and how many up or down? Then it rolls back from (7, 1) to (2, 3). What changes?"
Students usually count squares and say 5 right and 2 down, then 5 left and 2 up. Ask how they could get those numbers without the grid. Guide them to subtract: 7 - 2 = 5 and 1 - 3 = -2. Record the signs: right and up are positive, left and down are negative. The return trip gives 2 - 7 = -5 and 3 - 1 = 2. Tell students that these pairs of changes are the components of the robot's displacement vector.
Direct Instruction20 minutes
The rule. A vector drawn from initial point (x1, y1) to terminal point (x2, y2) has components ⟨x2 - x1, y2 - y1⟩. A short way to say it: head minus tail. Use Diagram 1 to show that the two components are the legs of the right triangle under the arrow, and that sliding the arrow so its tail sits at the origin puts its head exactly at the point whose coordinates are the components.
Name the points: decide which point is the initial point (tail) and which is the terminal point (head). The wording "from ... to ..." tells you.
Subtract x-coordinates: terminal x minus initial x gives the x-component.
Subtract y-coordinates: terminal y minus initial y gives the y-component. Put parentheses around negative coordinates before subtracting.
Write and read the result: write ⟨a, b⟩ and describe it: a positive a means right, negative means left; a positive b means up, negative means down.
Check with a sketch: the arrow from the tail should move the way the signs say.
Stress three consequences. Reversing the order gives the opposite vector: DC = -CD (Diagram 2). Two arrows in different places with the same components are equal vectors. The rule works the same way in three dimensions, one subtraction per coordinate.
Basic subtraction
Find the components of the vector from A(2, -1) to B(6, 4).
Equation: AB = ⟨6 - 2, 4 - (-1)⟩ = ⟨4, 5⟩
Negative components
Find the components of the vector from C(3, 5) to D(-1, 2).
Equation: CD = ⟨-1 - 3, 2 - 5⟩ = ⟨-4, -3⟩: 4 left and 3 down
Reversing the order
Now find the vector from D(-1, 2) to C(3, 5) and compare.
Equation: DC = ⟨3 - (-1), 5 - 2⟩ = ⟨4, 3⟩ = -CD
Working backward to the terminal point
A vector with components ⟨-2, 7⟩ has initial point (5, -3). Where is its terminal point?
Equation: x2 - 5 = -2 and y2 - (-3) = 7, so the terminal point is (3, 4)
Three dimensions
A drone moves from (1, 2, 0) to (4, 6, 5), in meters east, north and up. Find its displacement.
Equation: ⟨4 - 1, 6 - 2, 5 - 0⟩ = ⟨3, 4, 5⟩: 3 m east, 4 m north, 5 m up
Guided Practice10 minutes
Pairs complete the table. One partner computes and the other checks with a sketch on graph paper; they swap roles for each row.
Guided practice vectors
From
To
Components
Reads as
E(-2, -3)
F(1, 1)
⟨3, 4⟩
3 right, 4 up
G(0, 6)
H(-5, 6)
⟨-5, 0⟩
5 left, no vertical change
J(4, -2)
K(4, -9)
⟨0, -7⟩
straight down 7
M(-1.5, 2)
N(2.5, -0.5)
⟨4, -2.5⟩
4 right, 2.5 down
Listen for three errors: subtracting initial minus terminal, dropping the sign when subtracting a negative (writing -3 - -2 as -5), and putting the y-change first. Ask what a zero component means for the direction of the arrow.
Independent Practice15 minutes
A trail map uses coordinates in kilometers east and north of a parking lot. Students find the displacement for each leg of a hike, with the components and a sentence such as "5 km east and 3 km north":
Trailhead (1, -2) to waterfall (6, 1): ⟨5, 3⟩
Waterfall (6, 1) to summit (3, 9): ⟨-3, 8⟩
Summit (3, 9) back to the trailhead (1, -2): ⟨-2, -11⟩
A second group starts at the ranger station (-4, 0) and hikes a leg equal to the first leg. Where do they end? (1, 3)
Early finishers write the vector from the trailhead to the summit and explain why it is not the same as the first leg.
Closure5 minutes
Exit ticket: (1) Find the components of the vector from P(-4, 2) to Q(1, -6). (Answer: ⟨5, -8⟩.) (2) A classmate got ⟨-5, 8⟩ for the same vector. What did the classmate do? (3) A vector with components ⟨3, -1⟩ ends at the origin. Where does it start? (Answer: (-3, 1).)
Differentiation Strategies
For Struggling Students
Give a template: ⟨(head x) - (tail x), (head y) - (tail y)⟩, with boxes for each coordinate and parentheses printed around every box
Have students count squares on a grid first and then confirm the count by subtraction, so the two methods agree
Color-code: tail coordinates in one color, head coordinates in another, and the color order never changes
For Advanced Students
Prove with variables that the components of the vector from Q to P are the opposites of those from P to Q
Given three vertices of a parallelogram, find every possible fourth vertex by using equal vectors
Find the components of a displacement in three dimensions from GPS-style coordinates, then describe it in words
Assessment Guidance
What to Look For
Check that students always subtract in the same order, terminal minus initial, and can say why that order matters. Look for correct handling of negative coordinates, especially y2 - (-3). Ask students to read their components in words ("4 left, 3 down") and to match them with a sketch: a student who can do that is less likely to reverse the order. When students work backward, they should add the components to the initial point, and subtract them from the terminal point.
02
Classroom Activities
3 Activities
1
Component Treasure Hunt
20 minPairs
Pairs follow a chain of six clue cards across a coordinate grid. Each card gives a vector in component form, and the terminal point of each move is the initial point of the next. At the end, pairs check each leg in reverse by subtraction, so every leg is tested with the standard's rule.
Clue Cards (start at the origin)
Card 1: ⟨3, 2⟩; Card 2: ⟨-5, 1⟩; Card 3: ⟨4, -6⟩
Card 4: ⟨-1, 4⟩; Card 5: ⟨6, 3⟩; Card 6: ⟨-7, -4⟩
Answer key for the teacher: (3, 2), (-2, 3), (2, -3), (1, 1), (7, 4), and Card 6 returns to (0, 0)
Procedure
Partner A reads a card and plots the next point; Partner B draws the arrow and labels its tail and head
After all six moves, pairs list the seven points in order and find the components of each leg by subtracting coordinates, terminal minus initial
Every recomputed leg must match its card; a mismatch means a plotting or subtraction error to find and fix
Discussion Questions
Why did the hunt end at the starting point?
If Card 4 were read backward, from head to tail, what components would you get?
Modification for Distance Learning
Share the clue cards in a slide deck and have pairs plot on an online graphing tool, then paste a screenshot of their path and the subtraction table.
2
Head Minus Tail Error Hunt
15 minGroups of 3
Groups review six worked answers from an imaginary student. Some are right and some have one error. For each one, the group decides whether it is correct, names the error, and fixes it.
The Six Worked Answers
From P(2, 7) to Q(5, 1): student wrote ⟨-3, 6⟩ (order reversed; correct: ⟨3, -6⟩)
From R(-3, 4) to S(2, 9): student wrote ⟨-1, 5⟩ (added -3 instead of subtracting it; correct: ⟨5, 5⟩)
From U(1, -6) to V(4, -2): student wrote ⟨3, -8⟩ (sign error in -2 - (-6); correct: ⟨3, 4⟩)
From W(0, 3) to Z(-4, 0): student wrote ⟨-3, -4⟩ (components swapped; correct: ⟨-4, -3⟩)
From G(6, -1) to H(-2, -5): student wrote ⟨-8, -4⟩ (correct)
From K(-7, -2) to L(-1, 3): student wrote ⟨-8, 1⟩ (added coordinates; correct: ⟨6, 5⟩)
Procedure
Each student checks two answers alone, then explains them to the group
The group agrees on a verdict and writes the error type in a few words
Groups make a class poster of error types with one example each
Challenge Variation
Each group writes two new worked answers, one correct and one with a hidden error, and trades them with another group.
3
Radar Screen Displacements
20 minPairs
A printed radar screen shows four aircraft, with positions in kilometers east and north of a control tower, at 2:00 and at 2:01. Pairs find each plane's one-minute displacement vector and compare the vectors.
Radar Data
Plane A: (-12, 8) at 2:00, (-4, 2) at 2:01
Plane B: (5, -10) at 2:00, (13, -16) at 2:01
Plane C: (10, 10) at 2:00, (2, 16) at 2:01
Plane D: (-6, -3) at 2:00, (-6, 7) at 2:01
Procedure
Find each displacement as the later position minus the earlier one
Identify two planes whose displacements are equal vectors even though the planes are far apart
Identify two planes flying in exactly opposite directions
Describe Plane D's motion in words, and explain what its zero x-component means
Discussion Questions
Why must we subtract the 2:00 position from the 2:01 position, and not the other way?
Plane A moves 10 km each minute. Is that a realistic speed for a jet in kilometers per hour?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Components as Horizontal and Vertical Change
Vector v runs from A(2, -1) to B(6, 4). The dashed legs show the horizontal change 6 - 2 = 4 and the vertical change 4 - (-1) = 5, so v = ⟨4, 5⟩. The gray dashed arrow is the same vector with its initial point at the origin; its terminal point is (4, 5). Drawn to scale, 1 unit per square.
Diagram 2: Reversing the Order Reverses the Vector
The same two points give two different vectors. From C to D the components are ⟨-4, -3⟩ (left and down). From D to C they are ⟨4, 3⟩ (right and up), so DC = -CD. Both grids are drawn to scale.
04
Homework Assignment
~30 min
HSN.VM.A.2 Homework: Components from Initial and Terminal Points
Directions: Show each subtraction, with parentheses around negative coordinates. Write components in angle brackets, ⟨a, b⟩, and describe each vector in words (for example, "2 left and 5 up"). Sketch at least three of your vectors on graph paper.
Part 1: Components from Two Points (Problems 1-2)
Find the components of the vector from the first point to the second: (a) A(1, 4) to B(7, 9) (b) C(-2, 3) to D(-8, -1) (c) E(5, -4) to F(5, 2) (d) G(-3.5, 0) to H(1, -2.5)
Let M = (-1, -4) and N = (3, 2). Find the components of MN and of NM. How are the two answers related, and why?
Part 2: Working Backward and Equal Vectors (Problems 3-4)
(a) A vector with components ⟨-6, 5⟩ has initial point (2, -3). Find its terminal point. (b) A vector with components ⟨4, -9⟩ has terminal point (1, 1). Find its initial point. Check both answers by subtracting.
The points P(1, 1), Q(5, 2), R(7, 6) and S(3, 5) are the vertices of quadrilateral PQRS. Find the components of PQ, SR, QR and PS. Which pairs are equal vectors? What does that tell you about the shape?
Part 3: Components in Context (Problems 5-6)
A drone takes off from a landing pad at (4, -2, 0) and hovers at (-3, 10, 25), with coordinates in meters east, north and up from a fixed marker. Find the components of the drone's displacement and explain what each component means.
In a video game, a character jumps from (-15, 40) to (25, 10) on the map. Find the components of the jump. A second character standing at (60, -5) makes a jump with the same vector. Where does the second character land?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Subtraction Order
Terminal minus initial every time, with work shown
Order right but one sign error
Order reversed or coordinates added
Working Backward
Both missing points found and checked
One point correct or no check
Neither point correct
Interpretation
Every vector described in words and matched to a sketch
Descriptions given for some vectors
No descriptions or sketches
Reasoning
Clear explanations for opposite and equal vectors
Correct conclusion with a weak reason
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Answer each question, then read the explanation. Components are written in angle brackets, and bold letters are vectors. Your score updates as you go, and Reset quiz clears all answers for another attempt.
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
What are the components of the vector from (3, 1) to (8, 7)?
Answer: B
Terminal minus initial: ⟨8 - 3, 7 - 1⟩ = ⟨5, 6⟩. Choice A subtracts in the wrong order, choice C adds the coordinates, and choice D puts the y-change first.
Question 2 of 20 · Multiple Choice
What are the components of the vector from (-2, 5) to (4, -1)?
Answer: D
⟨4 - (-2), -1 - 5⟩ = ⟨6, -6⟩. Choice A adds the coordinates. Choice B subtracts initial minus terminal. Choice C loses the negative sign of the y-component.
Question 3 of 20 · Multiple Choice
A vector goes from P(x1, y1) to Q(x2, y2). Which expression gives the y-component of the vector from Q back to P?
Answer: A
For the vector from Q to P, Q is the initial point and P is the terminal point, so the y-component is (terminal y) - (initial y) = y1 - y2. Choice B is the y-component of the vector from P to Q. Choice C is an x-component, and choice D adds instead of subtracting.
Question 4 of 20 · Multiple Choice
What are the components of the vector from (-4, -3) to (-9, 2)?
Answer: C
⟨-9 - (-4), 2 - (-3)⟩ = ⟨-5, 5⟩. Choice A reverses the order. Choice B adds the coordinates instead of subtracting.
Question 5 of 20 · Multiple Choice
A vector with components ⟨-3, 8⟩ has initial point (6, -2). What is its terminal point?
Answer: A
Terminal = initial + components: (6 + (-3), -2 + 8) = (3, 6). Check: ⟨3 - 6, 6 - (-2)⟩ = ⟨-3, 8⟩. Choice B subtracts the components from the initial point, which finds a point that would be the initial point of a vector ending at (6, -2).
Question 6 of 20 · Multiple Choice
A vector with components ⟨2, -4⟩ has terminal point (0, 5). What is its initial point?
Answer: D
Initial = terminal - components: (0 - 2, 5 - (-4)) = (-2, 9). Check: ⟨0 - (-2), 5 - 9⟩ = ⟨2, -4⟩. Choice A adds the components to the terminal point, which moves in the wrong direction.
Question 7 of 20 · Multiple Choice
A vector from R(1, 1) to S has components ⟨0, -4⟩. Which describes the vector?
Answer: B
A zero x-component means no horizontal change, and -4 means 4 units down. So S = (1, -3) and the arrow points straight down. Choice D ignores the negative sign.
Question 8 of 20 · Multiple Choice
Which vector is equal to the vector from (0, 0) to (-3, 4)?
Answer: C
The target is ⟨-3, 4⟩. Choice C gives ⟨-1 - 2, 3 - (-1)⟩ = ⟨-3, 4⟩. Choice A is ⟨3, -4⟩, the opposite vector. Choice B is ⟨-3, -4⟩, and choice D is ⟨-4, 4⟩.
Question 9 of 20 · Multiple Choice
The vector from A to B has components ⟨7, -2⟩. What are the components of the vector from B to A?
Answer: A
Swapping the initial and terminal points changes the sign of every difference, so the components become ⟨-7, 2⟩. Choice B ignores the change in order, and choices C and D swap the components.
Question 10 of 20 · Multiple Choice
A drone flies from (2, -1, 4) to (5, 3, 0), in meters. What are the components of its displacement?
Answer: B
Subtract each coordinate: ⟨5 - 2, 3 - (-1), 0 - 4⟩ = ⟨3, 4, -4⟩, so the drone moves 4 m down. Choice A reverses the order. Choice D computes 3 - 1 instead of 3 - (-1).
Question 11 of 20 · Multiple Choice
A student finds the vector from (-3, 2) to (5, 6) and writes ⟨2, 4⟩. What went wrong?
Answer: D
The x-component is 5 - (-3) = 8, so the correct vector is ⟨8, 4⟩. The student's 2 comes from 5 + (-3). The y-component, 6 - 2 = 4, is right. Choice A would give ⟨-8, -4⟩.
Question 12 of 20 · Multiple Choice
A ship moves from (12, -8) to (-3, -8), in kilometers east and north of a port. What are the components of its displacement?
Answer: C
⟨-3 - 12, -8 - (-8)⟩ = ⟨-15, 0⟩: 15 km west with no north-south change. Choice A reverses the order. Choice D puts the change in the wrong component.
Question 13 of 20 · Multiple Choice
The vector v = ⟨4, -3⟩ is drawn with its initial point at the origin. What is its terminal point?
Answer: B
Terminal = (0 + 4, 0 + (-3)) = (4, -3). When a vector starts at the origin, its terminal point has the same coordinates as its components. Choice A is the terminal point of -v.
Question 14 of 20 · Multiple Choice
Points A(0, 0), B(5, 1) and C(9, 3) are three vertices of parallelogram ABCD. Find D so that AD = BC.
Answer: D
BC = ⟨9 - 5, 3 - 1⟩ = ⟨4, 2⟩. Since AD must equal it and A is the origin, D = (4, 2). Choice A adds the coordinates of B and C. Choice B uses the vector from C to B.
Question 15 of 20 · Short Answer
Find the components of the vector from J(-7, 3) to K(-2, -9), and describe the vector in words.
⟨-2 - (-7), -9 - 3⟩ = ⟨5, -12⟩: 5 units right and 12 units down.
Question 16 of 20 · Short Answer
A vector with components ⟨-5, 9⟩ ends at (-1, 4). Find its initial point, and check your answer.
Let P = (a, b) and Q = (c, d). Use the subtraction rule to show that the components of the vector from Q to P are the opposites of the components of the vector from P to Q.
PQ = ⟨c - a, d - b⟩ and QP = ⟨a - c, b - d⟩. Since a - c = -(c - a) and b - d = -(d - b), QP = -PQ: each component changes sign.
Question 18 of 20 · Short Answer
A hiker's GPS gives her position in kilometers east and north of a ranger station: (-1.2, 3.5) at 9:00 and (2.6, 0.9) at 10:00. Find the components of her displacement and describe it in words.
⟨2.6 - (-1.2), 0.9 - 3.5⟩ = ⟨3.8, -2.6⟩: she ended 3.8 km farther east and 2.6 km farther south than where she was at 9:00.
Question 19 of 20 · Short Answer
Is the vector from (1, -2) to (4, 3) equal to the vector from (-5, 0) to (-2, 5)? Justify your answer with components.
Yes. The first is ⟨4 - 1, 3 - (-2)⟩ = ⟨3, 5⟩ and the second is ⟨-2 - (-5), 5 - 0⟩ = ⟨3, 5⟩. Equal components mean the same magnitude and direction, even though the arrows are in different places.
Question 20 of 20 · Short Answer
The vector from (k, 2) to (7, -3) has x-component 10. Find k and the y-component.
7 - k = 10, so k = -3. The y-component is -3 - 2 = -5, so the vector is ⟨10, -5⟩.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSN.VM.A.2 mean?
HSN.VM.A.2 asks students to find the components of a vector from two points: subtract the coordinates of the initial point from the coordinates of the terminal point. For a vector from (1, 2) to (6, 0), the components are ⟨6 - 1, 0 - 2⟩ = ⟨5, -2⟩. The (+) in front of the standard marks it as advanced content for students headed to courses such as Precalculus or Calculus.
Is HSN.VM.A.2 part of Algebra 2 or Precalculus?
It is usually part of Precalculus, together with the other vector standards. Physics courses often teach the same skill for displacement, sometimes earlier. The subtraction itself only needs middle school integer skills, so it can be introduced whenever vectors are.
Why subtract the initial point from the terminal point and not the other way around?
Because a vector describes the change from where it starts to where it ends. "Ending position minus starting position" is the same idea as change in temperature or change in balance. Subtracting the other way describes the trip backward, which is the opposite vector.
What does a negative component mean?
A negative x-component means the vector points left (or west on a map), and a negative y-component means it points down (or south). A component of 0 means no change in that direction, so the arrow is horizontal or vertical. The sign carries direction; the size of the component carries the amount of change.
What notation is used for component form?
This page uses angle brackets, ⟨a, b⟩, so a vector is not confused with the point (a, b). Textbooks also use (a, b), a column with a on top of b, or ai + bj, where i and j are the unit vectors along the axes. All of these mean the same vector. Students should recognize each form.
What is the difference between a point and a vector in component form?
A point is a location. A vector is a change, with no fixed location. The vector ⟨2, 3⟩ can start anywhere; wherever it starts, it ends 2 units to the right and 3 units up. Only when the vector starts at the origin do its components match the coordinates of its terminal point.
Do equal vectors always have the same components?
Yes. Two vectors are equal when they have the same magnitude and direction, and that happens exactly when their components match. This is why component form is so useful: to test whether two arrows in different places are equal, subtract and compare, with no measuring.
Does this method work in three dimensions?
Yes. Subtract each coordinate the same way. For a vector from (x1, y1, z1) to (x2, y2, z2), the components are ⟨x2 - x1, y2 - y1, z2 - z1⟩. The standard does not limit students to the plane, and three-dimensional examples such as drone flights show why the rule matters.
What mistakes do students make when finding components?
Common ones are subtracting in the wrong order, mishandling a negative coordinate (writing 2 - (-3) as -1), adding coordinates instead of subtracting, and writing the y-change first. When working backward, students often subtract the components when they should add them. A quick sketch catches many of these errors, because the arrow must move the way the signs say.
How does HSN.VM.A.2 connect to later vector standards?
Component form is the tool for almost everything that follows. The magnitude of ⟨a, b⟩ is √(a² + b²), the distance formula from geometry. HSN.VM.A.3 uses components to solve velocity and force problems, and HSN.VM.B.4 and HSN.VM.B.5 add vectors and multiply them by scalars one component at a time.
07
Related Standards
6 standards
These standards connect to HSN.VM.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.NS.C.8Prerequisite
Graph points in all four quadrants and find distances between points with a shared coordinate
