A vector is a quantity that has both magnitude (size) and direction.
Examples:
Displacement: "5 miles east"
Velocity: "30 m/s at 45°"
Force: "10 N downward"
Notation
Vectors can be written as:
Bold: v
Arrow: v
📚 Practice Problems
1Problem 1easy
❓ Question:
Find the magnitude and direction angle of vector v=⟨3,4⟩.
💡 Show Solution
Solution:
Explain using:
⚠️ Common Mistakes: Vectors in the Plane
Avoid these 4 frequent errors
🌍 Real-World Applications: Vectors in the Plane
See how this math is used in the real world
📝 Worked Example: Related Rates — Expanding Circle
Problem:
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?
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.
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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.
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Component form: ⟨a,b⟩ or ⟨a,b⟩
Unit vector form: ai+bj
Component Form
A vector from the origin to point (a,b) is:
v=⟨a,b⟩
a is the horizontal component (x-component)
b is the vertical component (y-component)
Vector Between Two Points
The vector from point P1(x1,y1) to P2(x2,y2) is:
P1P2
Magnitude of a Vector
The magnitude (or length) of vector v=⟨a,b⟩ is:
∣v∣=a2+b2
This is the distance formula!
Direction of a Vector
The direction angleθ is measured counterclockwise from the positive x-axis:
tan(θ)=ab
θ=tan−1(ab)
⚠️ Be careful with quadrants when finding θ!
Vector Operations
Scalar Multiplication
If v=⟨a,b⟩ and k is a scalar:
kv=⟨ka,kb⟩
If k>0: same direction, scaled by k
If k<0: opposite direction, scaled by ∣k∣
Vector Addition
If u=⟨a1,b1⟩ and v=⟨a2,b2⟩:
u+v=⟨a1+a2,
Geometric interpretation: Tip-to-tail method or parallelogram rule
Vector Subtraction
u−v=⟨a1−a2,b1−b2⟩
Unit Vectors
A unit vector has magnitude 1.
The standard unit vectors are:
i=⟨1,0⟩ (horizontal)
j=⟨0,1⟩ (vertical)
Any vector can be written as: v=ai+bj
Finding a Unit Vector
To find a unit vector in the direction of v:
u=∣v∣v=∣v∣1v
Dot Product
The dot product of u=⟨a1,b1⟩ and v=⟨a2,b2⟩ is:
u⋅v=a1a2+b1b2
Properties
u⋅v=∣u∣∣v∣cos(θ) where θ is the angle between them
If u⋅v=0, the vectors are perpendicular
The dot product is a scalar, not a vector!
Find the magnitude:
∣v∣=a2+b2=32+42
=9+16=25=5
Find the direction angle:
tan(θ)=ab=34
θ=tan−1(34)
θ≈53.13°
Since both components are positive, the vector is in Quadrant I, so this angle is correct.
Answers:
Magnitude: 5
Direction: θ≈53.13° or 0.927 radians
2Problem 2easy
❓ Question:
Given vectors u=⟨3,−4⟩ and v=⟨−2,5⟩:
a) Find u+v
b) Find
c) Find the magnitude of
💡 Show Solution
Solution:
Part (a): Add components:
u
3Problem 3easy
❓ Question:
Given u=⟨2,−3⟩ and v=⟨−1,4⟩, find: (a) u+v, (b) 3u−2v
💡 Show Solution
Solution:
Part a)u+v
Add corresponding components:
u+v=
4Problem 4medium
❓ Question:
A vector w has magnitude 10 and makes an angle of 120° with the positive x-axis.
a) Write w in component form.
b) Find a unit vector in the direction of w.
💡 Show Solution
Solution:
Part (a): For a vector with magnitude r and angle θ:
w
5Problem 5easy
❓ Question:
Find the dot product of u=⟨5,2⟩ and v=⟨−3,4⟩. Are the vectors perpendicular?
Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.
=
⟨x2−
x1,y2−
y1⟩
b1
+
b2⟩
2u−3v
u
+
v=
⟨3,−4⟩+
⟨−2,5⟩=
⟨3+
(−2),−4+
5⟩=
⟨1,1⟩
Part (b): Scalar multiplication and subtraction:
2u=2⟨3,−4⟩=⟨6,−8⟩
3v=3⟨−2,5⟩=⟨−6,15⟩
2u−3v=⟨6,−8⟩−⟨−6,15⟩=⟨6−(−6),−8−15⟩=⟨12,−23⟩
Part (c): Magnitude formula: ∣u∣=u12+u22
∣u∣=32+(−4)2=9+16=25=5
⟨
2
,
−
3
⟩
+
⟨−1,4⟩
=⟨2+(−1),−3+4⟩
=⟨1,1⟩
Part b)3u−2v
Step 1: Scalar multiplication.
3u=3⟨2,−3⟩=⟨6,−9⟩
2v=2⟨−1,4⟩=⟨−2,8⟩
Step 2: Subtract.
3u−2v=⟨6,−9⟩−⟨−2,8⟩
=⟨6−(−2),−9−8⟩
=⟨8,−17⟩
Answers:
a) ⟨1,1⟩
b) ⟨8,−17⟩
=
⟨rcosθ,rsinθ⟩
w=⟨10cos120°,10sin120°⟩
cos120°=−21 and sin120°=23
w=⟨10(−21),10(23)⟩=⟨−5,53⟩
Part (b): A unit vector has magnitude 1. To find the unit vector in the direction of w:
w^=∣w∣w=10⟨−5,53⟩=⟨−21,23
Verify: ∣w^∣=(−21)2+(23)2=41+431=1 ✓
2
+
b1b2
=(5)(−3)+(2)(4)
=−15+8
=−7
Are they perpendicular?
Two vectors are perpendicular if and only if their dot product equals zero.
Since u⋅v=−7=0, the vectors are not perpendicular.