What is Vectors in the Plane?

Understanding vectors, vector operations, and magnitude and direction

How can I study Vectors in the Plane effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Vectors in the Plane study guide free?

Yes — all study notes, flashcards, and practice problems for Vectors in the Plane on Study Mondo are free to access. No account is needed.

What course covers Vectors in the Plane?

Vectors in the Plane is part of the AP Precalculus course on Study Mondo, specifically in the Functions Involving Parameters, Vectors, and Matrices section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Vectors in the Plane?

Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Vectors in the Plane

What is a Vector?

A vector is a quantity that has both (size) and .

❓ Question:

Find the magnitude and direction angle of vector $\mathbf{v} = \langle 3, 4 \rangle$ .

Solution:

Find the magnitude:

$|\mathbf{v}| = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2}$

$= \sqrt{9 + 16} = \sqrt{25} = 5$

Find the direction angle:

$\tan(\theta) = \frac{b}{a} = \frac{4}{3}$

$\theta = \tan^{-1}\left(\frac{4}{3}\right)$

$\theta \approx 53.13°$

Since both components are positive, the vector is in Quadrant I, so this angle is correct.

Answers:

Magnitude: $5$
Direction: $\theta \approx 53.13°$ or $0.927$ radians

❓ Question:

Given vectors $\vec{u} = \langle 3, -4 \rangle$ and :

❓ Question:

Given $\mathbf{u} = \langle 2, -3 \rangle$ and $\mathbf{v} = \langle -1, 4 \rangle$ , find: (a) , (b)

❓ Question:

A vector $\vec{w}$ has magnitude 10 and makes an angle of $120°$ with the positive -axis.

❓ Question:

Find the dot product of $\mathbf{u} = \langle 5, 2 \rangle$ and $\mathbf{v} = \langle -3, 4 \rangle$ . Are the vectors perpendicular?

A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of $2$ cm/s. How fast is the area of the circle increasing when the radius is $10$ cm?

Solution:

Part (a): Add components:

$\vec{u} + \vec{v} = \langle 3, -4 \rangle + \langle -2, 5 \rangle = \langle 3 + (-2), -4 + 5 \rangle = \langle 1, 1 \rangle$

Part (b): Scalar multiplication and subtraction:

$2\vec{u} = 2\langle 3, -4 \rangle = \langle 6, -8 \rangle$

$3\vec{v} = 3\langle -2, 5 \rangle = \langle -6, 15 \rangle$

$2\vec{u} - 3\vec{v} = \langle 6, -8 \rangle - \langle -6, 15 \rangle = \langle 6-(-6), -8-15 \rangle = \langle 12, -23 \rangle$

Part (c): Magnitude formula: $|\vec{u}| = \sqrt{u_1^2 + u_2^2}$

$|\vec{u}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Solution:

Part a) $\mathbf{u} + \mathbf{v}$

Add corresponding components: $\mathbf{u} + \mathbf{v} = \langle 2, -3 \rangle + \langle -1, 4 \rangle$

$= \langle 2 + (-1), -3 + 4 \rangle$

$= \langle 1, 1 \rangle$

Part b) $3\mathbf{u} - 2\mathbf{v}$

Step 1: Scalar multiplication. $3\mathbf{u} = 3\langle 2, -3 \rangle = \langle 6, -9 \rangle$

$2\mathbf{v} = 2\langle -1, 4 \rangle = \langle -2, 8 \rangle$

Step 2: Subtract. $3\mathbf{u} - 2\mathbf{v} = \langle 6, -9 \rangle - \langle -2, 8 \rangle$

$= \langle 6 - (-2), -9 - 8 \rangle$

$= \langle 8, -17 \rangle$

Answers:

a) $\langle 1, 1 \rangle$
b) $\langle 8, -17 \rangle$

Solution:

Part (a): For a vector with magnitude $r$ and angle $\theta$ :

$\vec{w} = \langle r\cos\theta, r\sin\theta \rangle$

$\vec{w} = \langle 10\cos 120°, 10\sin 120° \rangle$

$\cos 120° = -\frac{1}{2}$ and $\sin 120° = \frac{\sqrt{3}}{2}$

$\vec{w} = \left\langle 10\left(-\frac{1}{2}\right), 10\left(\frac{\sqrt{3}}{2}\right) \right\rangle = \langle -5, 5\sqrt{3} \rangle$

Part (b): A unit vector has magnitude 1. To find the unit vector in the direction of $\vec{w}$ :

$\hat{w} = \frac{\vec{w}}{|\vec{w}|} = \frac{\langle -5, 5\sqrt{3} \rangle}{10} = \left\langle -\frac{1}{2}, \frac{\sqrt{3}}{2} \right\rangle$

Verify: $\left|\hat{w}\right| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1$ ✓

Solution:

Find the dot product:

$\mathbf{u} \cdot \mathbf{v} = a_1a_2 + b_1b_2$

$= (5)(-3) + (2)(4)$

$= -15 + 8$

$= -7$

Are they perpendicular?

Two vectors are perpendicular if and only if their dot product equals zero.

Since $\mathbf{u} \cdot \mathbf{v} = -7 \neq 0$ , the vectors are not perpendicular.

Answers:

Dot product: $-7$
The vectors are not perpendicular

Vectors in the Plane

Try the Interactive Version!

Vectors in the Plane

What is a Vector?

Notation

Component Form

Vector Between Two Points

Magnitude of a Vector

Direction of a Vector

Vector Operations

Scalar Multiplication

Vector Addition

Vector Subtraction

Unit Vectors

Finding a Unit Vector

Dot Product

Properties

📚 Practice Problems

1Problem 1easy

❓ Question:

2Problem 2easy

❓ Question:

3Problem 3easy

❓ Question:

4Problem 4medium

❓ Question:

5Problem 5easy

❓ Question:

⚠️ Common Mistakes: Vectors in the Plane

🌍 Real-World Applications: Vectors in the Plane

📝 Worked Example: Related Rates — Expanding Circle

Practice Lab

Practice with Flashcards

Browse All Topics

📌 Related Topics in Functions Involving Parameters, Vectors, and Matrices

❓ Frequently Asked Questions