Adding and Subtracting Vectors: Examples

Adding and subtracting vectors is done one component at a time: $\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$ , and subtraction works the same way. On a graph, place the tail of the second vector at the tip of the first. The arrow from start to finish is the sum, called the resultant.

Before this lesson: Magnitude and Direction of a Vector and Translations

What it means to add vectors

Think of a vector as a trip. Walk 3 blocks east and 1 block north, then walk 1 block east and 4 blocks north. In total you moved 4 blocks east and 5 blocks north. That total trip is the sum of the two vectors, also called the resultant.

On a graph, this is the tip-to-tail method. Draw the first vector, start the second vector at its tip, and draw an arrow from the first tail to the last tip.

Subtracting works the same way, because $\vec{u} - \vec{v} = \vec{u} + (-\vec{v})$ . The vector $-\vec{v}$ has the same length as $\vec{v}$ but points the opposite way.

Left graph: u = ⟨3, 1⟩ starts at the origin and v = ⟨1, 4⟩ starts at the tip of u. The resultant u + v = ⟨4, 5⟩ runs from the origin to (4, 5). Right graph: u = ⟨3, 1⟩, then −v = ⟨−1, −4⟩ starts at the tip of u, so u − v = ⟨2, −3⟩ runs from the origin to (2, −3). — Tip to tail: the dashed resultant goes from the first tail to the last tip. To subtract v, add its opposite.

Rules for adding, subtracting and scaling vectors

For $\vec{u} = \langle a, b \rangle$ , $\vec{v} = \langle c, d \rangle$ and any real number k:

Operation	Rule	What it does to the arrow
Addition	$\vec{u} + \vec{v} = \langle a + c, b + d \rangle$	Tip to tail; the resultant joins the first tail to the last tip
Subtraction	$\vec{u} - \vec{v} = \langle a - c, b - d \rangle$	Adds the opposite of v
Scalar multiplication	$k\vec{u} = \langle ka, kb \rangle$	Stretches or shrinks the arrow by k; a negative k also flips it
Opposite	$-\vec{u} = \langle -a, -b \rangle$	Same length, opposite direction

A scalar is a plain number. Multiplying by a scalar changes the length, so $\|k\vec{u}\| = |k|\,\|\vec{u}\|$ . Vector addition is also commutative, so $\vec{u} + \vec{v} = \vec{v} + \vec{u}$ .

Adding and subtracting vectors examples

Example 1: Add two vectors

Find $\vec{u} + \vec{v}$ for $\vec{u} = \langle 3, 1 \rangle$ and $\vec{v} = \langle 1, 4 \rangle$ .

\vec{u} + \vec{v} = \langle 3 + 1, 1 + 4 \rangle

Add the x-components and add the y-components.

\vec{u} + \vec{v} = \langle 4, 5 \rangle

Add.

Answer: $\langle 4, 5 \rangle$ . This is the left graph above: 3 right and 1 up, then 1 right and 4 up, ends at (4, 5).

Example 2: Subtract two vectors

Find $\vec{u} - \vec{v}$ for $\vec{u} = \langle 2, -5 \rangle$ and $\vec{v} = \langle -4, 3 \rangle$ .

\vec{u} - \vec{v} = \langle 2 - (-4), -5 - 3 \rangle

Subtract each component of v from the matching component of u.

\vec{u} - \vec{v} = \langle 2 + 4, -5 - 3 \rangle

Subtracting −4 is the same as adding 4 (see subtracting integers).

\vec{u} - \vec{v} = \langle 6, -8 \rangle

Simplify each component.

Answer: $\langle 6, -8 \rangle$ .

Example 3: Scalar multiples and a combination

Find $3\vec{u} - 2\vec{v}$ for $\vec{u} = \langle 4, -1 \rangle$ and $\vec{v} = \langle -2, 5 \rangle$ .

3\vec{u} = \langle 3(4), 3(-1) \rangle

Multiply each component of u by 3.

3\vec{u} = \langle 12, -3 \rangle

Multiply (see multiplying integers).

2\vec{v} = \langle 2(-2), 2(5) \rangle

Multiply each component of v by 2.

2\vec{v} = \langle -4, 10 \rangle

Multiply.

3\vec{u} - 2\vec{v} = \langle 12 - (-4), -3 - 10 \rangle

Subtract the components.

3\vec{u} - 2\vec{v} = \langle 16, -13 \rangle

Simplify each component.

Answer: $\langle 16, -13 \rangle$ .

Example 4: Add vectors given by magnitude and direction

Vector a has magnitude 10 and direction angle 30°. Vector b has magnitude 6 and direction angle 120°. Find the magnitude and direction of $\vec{a} + \vec{b}$ .

\vec{a} = \langle 10\cos 30^\circ, 10\sin 30^\circ \rangle

Write a in component form (see magnitude and direction of a vector).

\vec{a} \approx \langle 8.660, 5 \rangle

Use $\cos 30^\circ \approx 0.8660$ and $\sin 30^\circ = 0.5$ .

\vec{b} = \langle 6\cos 120^\circ, 6\sin 120^\circ \rangle

Write b in component form.

\vec{b} \approx \langle -3, 5.196 \rangle

Use $\cos 120^\circ = -0.5$ and $\sin 120^\circ \approx 0.8660$ (see the unit circle).

\vec{a} + \vec{b} \approx \langle 8.660 + (-3), 5 + 5.196 \rangle

Add the components.

\vec{a} + \vec{b} \approx \langle 5.660, 10.196 \rangle

Simplify.

\|\vec{a} + \vec{b}\| \approx \sqrt{5.660^2 + 10.196^2}

Use the magnitude formula.

\|\vec{a} + \vec{b}\| \approx \sqrt{135.99}

Square each component and add.

\|\vec{a} + \vec{b}\| \approx 11.66

Take the square root and round to the nearest hundredth.

\theta \approx \tan^{-1}\left(\frac{10.196}{5.660}\right)

Both components are positive, so the sum is in Quadrant I.

\theta \approx 61.0^\circ

Use a calculator in degree mode.

Answer: $\vec{a} + \vec{b}$ has magnitude about 11.66 and direction angle about 61.0°. You cannot just add the lengths (16) or the angles (150°).

Example 5: Word problem with a delivery drone

A drone flies from its base along $\langle 4, 3 \rangle$ , then along $\langle -6, 2 \rangle$ , in kilometers (east and north are positive). Where does it end up, how far is that from the base, and what vector takes it straight home?

\vec{r} = \langle 4 + (-6), 3 + 2 \rangle

Add the two legs to get the resultant r.

\vec{r} = \langle -2, 5 \rangle

Simplify. The drone is 2 km west and 5 km north of the base.

\|\vec{r}\| = \sqrt{(-2)^2 + 5^2}

Use the magnitude formula.

\|\vec{r}\| = \sqrt{4 + 25}

Square each component.

\|\vec{r}\| = \sqrt{29}

Add.

\|\vec{r}\| \approx 5.39

Round to the nearest hundredth.

-\vec{r} = \langle 2, -5 \rangle

The trip home is the opposite of r.

Answer: the drone ends at (−2, 5), about 5.39 km from the base. Flying $\langle 2, -5 \rangle$ brings it straight home.

Common mistakes

Mistake	Example of the error	Fix
Adding the lengths instead of the components	$\\|\langle 3, 0 \rangle + \langle 0, 4 \rangle\\| = 3 + 4 = 7$	Add first: $\langle 3, 4 \rangle$ , whose length is $\sqrt{3^2 + 4^2} = 5$ .
Adding the direction angles	30° and 120° give a sum at 150°	Change both to components, add, then find the new angle (61.0° in Example 4).
Dropping a negative sign when subtracting	$\langle 2, -5 \rangle - \langle -4, 3 \rangle = \langle -2, -2 \rangle$	Subtract each component: $\langle 2 + 4, -5 - 3 \rangle = \langle 6, -8 \rangle$ .
Scaling only one component	$3\langle 2, 5 \rangle = \langle 6, 5 \rangle$	Multiply both: $\langle 6, 15 \rangle$ .
Drawing the sum from tip to tip	Arrows u and v drawn from the same point, with u + v drawn from the tip of v to the tip of u	That arrow is u − v. For u + v, place v at the tip of u, or draw the long diagonal of the parallelogram.
Switching the order of subtraction	Finding v − u when asked for u − v	The two answers are opposites, so the order matters.

Related methods

The parallelogram method

Draw u and v from the same starting point and complete the parallelogram. The diagonal from the shared tail is $\vec{u} + \vec{v}$ . The other diagonal, from the tip of v to the tip of u, is $\vec{u} - \vec{v}$ .

Unit vectors

A unit vector has length 1. To get the unit vector in the direction of v, divide v by its magnitude: $\frac{\vec{v}}{\|\vec{v}\|}$ . For $\langle 3, 4 \rangle$ , the unit vector is $\left\langle \frac{3}{5}, \frac{4}{5} \right\rangle$ .

i and j form

With $\mathbf{i} = \langle 1, 0 \rangle$ and $\mathbf{j} = \langle 0, 1 \rangle$ , add like terms: $(3\mathbf{i} + \mathbf{j}) + (\mathbf{i} + 4\mathbf{j}) = 4\mathbf{i} + 5\mathbf{j}$ . It works just like combining like terms.

Practice problems

Find $\langle 5, -2 \rangle + \langle -1, 7 \rangle$ .
Find $\langle 3, 8 \rangle - \langle 6, -2 \rangle$ .
Find $4\langle -2, 3 \rangle$ .
Find $-\frac{1}{2}\langle 6, -10 \rangle$ .
For $\vec{u} = \langle 1, -4 \rangle$ and $\vec{v} = \langle 3, 5 \rangle$ , find $2\vec{u} + \vec{v}$ .
For $\vec{u} = \langle 7, 2 \rangle$ and $\vec{v} = \langle 2, -1 \rangle$ , find $\vec{u} - 3\vec{v}$ .
For $\vec{u} = \langle 2, 6 \rangle$ and $\vec{v} = \langle 4, 2 \rangle$ , find $\|\vec{u} + \vec{v}\|$ .
Find the unit vector in the direction of $\langle -5, 12 \rangle$ .
Find the vector w so that $\langle 3, -1 \rangle + \vec{w} = \langle -2, 6 \rangle$ .
Two ropes pull on a crate. One pulls with 20 newtons at 0° and the other with 15 newtons at 60°. Find the magnitude and direction of the total force.

Show answers

\langle 5, -2 \rangle + \langle -1, 7 \rangle = \langle 5 + (-1), -2 + 7 \rangle

Add the matching components.

\langle 5, -2 \rangle + \langle -1, 7 \rangle = \langle 4, 5 \rangle

Simplify.

Answer: $\langle 4, 5 \rangle$

\langle 3, 8 \rangle - \langle 6, -2 \rangle = \langle 3 - 6, 8 - (-2) \rangle

Subtract the matching components.

\langle 3, 8 \rangle - \langle 6, -2 \rangle = \langle -3, 10 \rangle

Simplify. Subtracting −2 adds 2.

Answer: $\langle -3, 10 \rangle$

4\langle -2, 3 \rangle = \langle 4(-2), 4(3) \rangle

Multiply each component by 4.

4\langle -2, 3 \rangle = \langle -8, 12 \rangle

Multiply.

Answer: $\langle -8, 12 \rangle$

-\frac{1}{2}\langle 6, -10 \rangle = \left\langle -\frac{1}{2}(6), -\frac{1}{2}(-10) \right\rangle

Multiply each component by $-\frac{1}{2}$ (see multiplying fractions).

-\frac{1}{2}\langle 6, -10 \rangle = \langle -3, 5 \rangle

Multiply. The arrow is half as long and points the opposite way.

Answer: $\langle -3, 5 \rangle$

2\vec{u} = \langle 2, -8 \rangle

Multiply each component of u by 2.

2\vec{u} + \vec{v} = \langle 2 + 3, -8 + 5 \rangle

Add v.

2\vec{u} + \vec{v} = \langle 5, -3 \rangle

Simplify.

Answer: $\langle 5, -3 \rangle$

3\vec{v} = \langle 6, -3 \rangle

Multiply each component of v by 3.

\vec{u} - 3\vec{v} = \langle 7 - 6, 2 - (-3) \rangle

Subtract from u.

\vec{u} - 3\vec{v} = \langle 1, 5 \rangle

Simplify.

Answer: $\langle 1, 5 \rangle$

\vec{u} + \vec{v} = \langle 2 + 4, 6 + 2 \rangle

Add the components first.

\vec{u} + \vec{v} = \langle 6, 8 \rangle

Simplify.

\|\vec{u} + \vec{v}\| = \sqrt{6^2 + 8^2}

Use the magnitude formula.

\|\vec{u} + \vec{v}\| = \sqrt{100}

Square and add: 36 + 64 = 100.

\|\vec{u} + \vec{v}\| = 10

Take the square root.

Answer: 10

\|\vec{v}\| = \sqrt{(-5)^2 + 12^2}

Find the magnitude first.

\|\vec{v}\| = \sqrt{169}

Square and add: 25 + 144 = 169.

\|\vec{v}\| = 13

Take the square root.

\frac{\vec{v}}{\|\vec{v}\|} = \frac{1}{13}\langle -5, 12 \rangle

Divide the vector by its magnitude.

\frac{\vec{v}}{\|\vec{v}\|} = \left\langle -\frac{5}{13}, \frac{12}{13} \right\rangle

Divide each component by 13.

Answer: $\left\langle -\frac{5}{13}, \frac{12}{13} \right\rangle$ . Check: $\frac{25}{169} + \frac{144}{169} = 1$ , so its length is 1.

\vec{w} = \langle -2, 6 \rangle - \langle 3, -1 \rangle

Subtract $\langle 3, -1 \rangle$ from both sides.

\vec{w} = \langle -2 - 3, 6 - (-1) \rangle

Subtract the components.

\vec{w} = \langle -5, 7 \rangle

Simplify.

Answer: $\langle -5, 7 \rangle$

10.

\vec{p} = \langle 20, 0 \rangle

The first rope pulls 20 N along the positive x-axis.

\vec{q} = \langle 15\cos 60^\circ, 15\sin 60^\circ \rangle

Write the second force in component form.

\vec{q} \approx \langle 7.5, 12.990 \rangle

Use $\cos 60^\circ = 0.5$ and $\sin 60^\circ \approx 0.8660$ .

\vec{p} + \vec{q} \approx \langle 27.5, 12.990 \rangle

Add the components.

\|\vec{p} + \vec{q}\| \approx \sqrt{27.5^2 + 12.990^2}

Use the magnitude formula.

\|\vec{p} + \vec{q}\| \approx \sqrt{924.99}

Square each component and add.

\|\vec{p} + \vec{q}\| \approx 30.41

Take the square root.

\theta \approx \tan^{-1}\left(\frac{12.990}{27.5}\right)

Both components are positive, so the angle is in Quadrant I.

\theta \approx 25.3^\circ

Use a calculator in degree mode.

Answer: about 30.41 N at a direction angle of about 25.3°.

Adding and subtracting vectors FAQ

How do you add two vectors?

Add the x-components and add the y-components: $\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$ . On a graph, place them tip to tail and draw the resultant from the first tail to the last tip.

How do you subtract vectors?

Subtract the matching components, or add the opposite: $\vec{u} - \vec{v} = \vec{u} + (-\vec{v})$ . Order matters, since $\vec{v} - \vec{u}$ is the opposite of $\vec{u} - \vec{v}$ .

What is a resultant vector?

It is the single vector equal to the sum of two or more vectors. It has the same effect as all of them together, like one trip that replaces several legs.

How do you find a unit vector?

Divide the vector by its magnitude. The result points the same way and has length 1, as in $\frac{1}{13}\langle -5, 12 \rangle$ .

Summary

Add or subtract vectors one component at a time.
On a graph, add tip to tail; the resultant runs from the first tail to the last tip.
Subtracting v is adding $-\vec{v}$ , the same length pointing the opposite way.
A scalar k multiplies every component and stretches the vector by |k|; a negative k flips it.
To add vectors given by magnitude and direction, change them to components first.

Previous lesson: Magnitude and Direction of a Vector

Next lesson: Dot Product

Back to the Precalculus Index