Practice Vector Addition, Subtraction, and Scalar Multiplication in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Vectors are added and subtracted component by component. Scalar multiplication multiplies each component of a vector by a number. If u=⟨u1,u2⟩\mathbf{u} = \langle u_1, u_2 \rangle and v=⟨v1,v2⟩\mathbf{v} = \langle v_1, v_2 \rangle, then u+v=⟨u1+v1,u2+v2⟩\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle and ku=⟨ku1,ku2⟩k\mathbf{u} = \langle ku_1, ku_2 \rangle.

Vectors are arrows with direction and magnitude. Adding two vectors is like walking along the first arrow, then continuing along the second—you end up at the tip of the combined arrow (tip-to-tail method). Scalar multiplication stretches or shrinks the arrow: 2v2\mathbf{v} is twice as long in the same direction, while −v-\mathbf{v} points the opposite way.

Showing a random 20 of 50 problems.

Example 1

medium
True or false: k(u+v)=ku+kvk(\mathbf{u} + \mathbf{v}) = k\mathbf{u} + k\mathbf{v} for any scalar kk.

Example 2

challenge
Points A(1,2)A(1, 2), B(5,4)B(5, 4), C(7,8)C(7, 8) are three vertices of a parallelogram ABCDABCD (in order). Find DD using vectors.

Example 3

hard
If u=⟨2,−1,3⟩\mathbf{u} = \langle 2, -1, 3 \rangle, find a vector parallel to u\mathbf{u} with magnitude 77.

Example 4

easy
Compute 2⟨1,1⟩+⟨0,3⟩2\langle 1, 1 \rangle + \langle 0, 3 \rangle.

Example 5

easy
Compute 4⟨−1,3⟩4\langle -1, 3 \rangle.

Example 6

medium
Why is ⟨1,2⟩+3\langle 1, 2 \rangle + 3 undefined?

Example 7

easy
Compute ⟨1,2⟩+⟨3,4⟩\langle 1, 2 \rangle + \langle 3, 4 \rangle as a component sum.

Example 8

medium
Show that u=⟨4,6⟩\mathbf{u} = \langle 4, 6 \rangle and v=⟨2,3⟩\mathbf{v} = \langle 2, 3 \rangle are parallel (one is a scalar multiple of the other).

Example 9

medium
Find the vector x\mathbf{x} that satisfies x+⟨2,−5⟩=⟨−1,6⟩\mathbf{x} + \langle 2, -5 \rangle = \langle -1, 6 \rangle.

Example 10

easy
Compute ⟨4,1⟩−⟨1,5⟩\langle 4, 1 \rangle - \langle 1, 5 \rangle.

Example 11

easy
What is the sum u+(−u)\mathbf{u} + (-\mathbf{u}) for any vector u\mathbf{u}?

Example 12

medium
Given u=⟨1,2,3⟩\mathbf{u} = \langle 1, 2, 3 \rangle and v=⟨4,−1,0⟩\mathbf{v} = \langle 4, -1, 0 \rangle, find u+2v\mathbf{u} + 2\mathbf{v}.

Example 13

medium
Fill in: ⟨a,b⟩+⟨c,d⟩=⟨ ‾ , ‾ ⟩\langle a, b \rangle + \langle c, d \rangle = \langle\, \underline{\quad}\, , \, \underline{\quad}\, \rangle.

Example 14

medium
Forces F1=⟨6,0⟩\mathbf{F}_1 = \langle 6, 0 \rangle N and F2=⟨−2,5⟩\mathbf{F}_2 = \langle -2, 5 \rangle N act on a particle. Find the net force.

Example 15

challenge
Are ⟨1,2⟩\langle 1, 2 \rangle and ⟨2,4⟩\langle 2, 4 \rangle linearly independent? Explain.

Example 16

medium
Find v⃗\vec{v} if 2v⃗=⟨8,−6⟩2\vec{v} = \langle 8, -6 \rangle.

Example 17

medium
Find the midpoint of segment from A(2,6)A(2, 6) to B(8,−2)B(8, -2) using the vector formula 12(OA⃗+OB⃗)\tfrac{1}{2}(\vec{OA} + \vec{OB}).

Example 18

medium
Find scalars ss and tt so that s⟨1,0⟩+t⟨0,1⟩=⟨7,−4⟩s\langle 1, 0 \rangle + t\langle 0, 1 \rangle = \langle 7, -4 \rangle.

Example 19

challenge
Find scalars a,ba, b with a⟨1,1⟩+b⟨1,−1⟩=⟨4,2⟩a\langle 1, 1 \rangle + b\langle 1, -1 \rangle = \langle 4, 2 \rangle.

Example 20

challenge
Vectors a\mathbf{a} and b\mathbf{b} satisfy 2a+3b=⟨1,0⟩2\mathbf{a} + 3\mathbf{b} = \langle 1, 0 \rangle and a−b=⟨0,1⟩\mathbf{a} - \mathbf{b} = \langle 0, 1 \rangle. Find a\mathbf{a}.
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