Two-Dimensional Vectors
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Vector Notation
Forms of Vectors:
Component Form:
\[ \vec{v} = \langle a, b \rangle \]
Unit Vector Form:
\[ \vec{v} = a\vec{i} + b\vec{j} \]
where \( \vec{i} = \langle 1, 0 \rangle \) and \( \vec{j} = \langle 0, 1 \rangle \)
2. Component Form of a Vector
Formula:
Given initial point \( P_1(x_1, y_1) \) and terminal point \( P_2(x_2, y_2) \):
\[ \vec{v} = \langle x_2 - x_1, y_2 - y_1 \rangle \]
The components represent horizontal and vertical displacement
Example:
Find component form: Initial point (2, 3), Terminal point (5, 7)
\( \vec{v} = \langle 5 - 2, 7 - 3 \rangle = \langle 3, 4 \rangle \)
3. Magnitude of a Vector
Formula:
For vector \( \vec{v} = \langle a, b \rangle \):
\[ |\vec{v}| = \|\vec{v}\| = \sqrt{a^2 + b^2} \]
The magnitude is the length of the vector (always non-negative)
Examples:
Find magnitude: \( \vec{v} = \langle 3, 4 \rangle \)
\( |\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
Find magnitude: \( \vec{v} = \langle -5, 12 \rangle \)
\( |\vec{v}| = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \)
4. Direction Angle of a Vector
Formula:
For vector \( \vec{v} = \langle a, b \rangle \), the direction angle \( \theta \) is:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]
⚠️ Must adjust for the correct quadrant!
Quadrant Adjustments:
| Quadrant | Condition | Angle Formula |
|---|---|---|
| I | \( a > 0, b > 0 \) | \( \theta = \tan^{-1}(b/a) \) |
| II | \( a < 0, b > 0 \) | \( \theta = 180° + \tan^{-1}(b/a) \) |
| III | \( a < 0, b < 0 \) | \( \theta = 180° + \tan^{-1}(b/a) \) |
| IV | \( a > 0, b < 0 \) | \( \theta = 360° + \tan^{-1}(b/a) \) |
5. Component Form from Magnitude and Direction
Formula:
Given magnitude \( |\vec{v}| = r \) and direction angle \( \theta \):
\[ \vec{v} = \langle r\cos\theta, r\sin\theta \rangle \]
or \( \vec{v} = r\cos\theta \vec{i} + r\sin\theta \vec{j} \)
Example:
Find component form: \( |\vec{v}| = 10 \), \( \theta = 60° \)
\( a = 10\cos(60°) = 10 \cdot \frac{1}{2} = 5 \)
\( b = 10\sin(60°) = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3} \)
\( \vec{v} = \langle 5, 5\sqrt{3} \rangle \)
6. Add Vectors
Formula:
Add components separately:
\[ \vec{u} + \vec{v} = \langle a_1, b_1 \rangle + \langle a_2, b_2 \rangle = \langle a_1 + a_2, b_1 + b_2 \rangle \]
Graphical Methods:
Triangle Method (Head-to-Tail):
1. Draw first vector \( \vec{u} \)
2. Place tail of \( \vec{v} \) at head of \( \vec{u} \)
3. Resultant goes from tail of \( \vec{u} \) to head of \( \vec{v} \)
Parallelogram Method:
1. Draw both vectors from the same starting point
2. Complete the parallelogram
3. Resultant is the diagonal from the common starting point
Example:
Add: \( \vec{u} = \langle 2, 3 \rangle \) and \( \vec{v} = \langle 4, -1 \rangle \)
\( \vec{u} + \vec{v} = \langle 2 + 4, 3 + (-1) \rangle = \langle 6, 2 \rangle \)
7. Subtract Vectors
Formula:
\[ \vec{u} - \vec{v} = \langle a_1, b_1 \rangle - \langle a_2, b_2 \rangle = \langle a_1 - a_2, b_1 - b_2 \rangle \]
Or equivalently: \( \vec{u} - \vec{v} = \vec{u} + (-\vec{v}) \)
Example:
Subtract: \( \vec{u} = \langle 5, 7 \rangle \) and \( \vec{v} = \langle 2, 3 \rangle \)
\( \vec{u} - \vec{v} = \langle 5 - 2, 7 - 3 \rangle = \langle 3, 4 \rangle \)
8. Scalar Multiplication
Formula:
Multiply each component by the scalar:
\[ k\vec{v} = k\langle a, b \rangle = \langle ka, kb \rangle \]
Properties:
• Magnitude: \( |k\vec{v}| = |k| \cdot |\vec{v}| \)
• If \( k > 0 \): direction stays the same
• If \( k < 0 \): direction reverses (opposite)
• If \( k = 0 \): result is zero vector
Example:
Find: \( 3\vec{v} \) where \( \vec{v} = \langle 2, -4 \rangle \)
\( 3\vec{v} = 3\langle 2, -4 \rangle = \langle 6, -12 \rangle \)
Original magnitude: \( |\vec{v}| = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \)
New magnitude: \( |3\vec{v}| = \sqrt{36 + 144} = \sqrt{180} = 6\sqrt{5} = 3(2\sqrt{5}) \)
9. Unit Vectors
Definition:
A unit vector has magnitude 1. To find the unit vector in the direction of \( \vec{v} \):
\[ \hat{v} = \frac{\vec{v}}{|\vec{v}|} = \frac{1}{|\vec{v}|}\vec{v} \]
Standard Unit Vectors:
\[ \vec{i} = \langle 1, 0 \rangle \quad \vec{j} = \langle 0, 1 \rangle \]
Example:
Find unit vector: \( \vec{v} = \langle 3, 4 \rangle \)
Magnitude: \( |\vec{v}| = \sqrt{9 + 16} = 5 \)
Unit vector: \( \hat{v} = \frac{1}{5}\langle 3, 4 \rangle = \langle \frac{3}{5}, \frac{4}{5} \rangle \)
Verify: \( |\hat{v}| = \sqrt{\frac{9}{25} + \frac{16}{25}} = \sqrt{\frac{25}{25}} = 1 \) ✓
10. Linear Combinations of Vectors
Definition:
A linear combination uses scalar multiplication and addition:
\[ \vec{w} = c_1\vec{v}_1 + c_2\vec{v}_2 \]
where \( c_1 \) and \( c_2 \) are scalars
Example:
Find: \( 2\vec{u} - 3\vec{v} \) where \( \vec{u} = \langle 4, 1 \rangle \) and \( \vec{v} = \langle 2, -3 \rangle \)
\( 2\vec{u} = 2\langle 4, 1 \rangle = \langle 8, 2 \rangle \)
\( 3\vec{v} = 3\langle 2, -3 \rangle = \langle 6, -9 \rangle \)
\( 2\vec{u} - 3\vec{v} = \langle 8, 2 \rangle - \langle 6, -9 \rangle = \langle 2, 11 \rangle \)
11. Quick Reference Summary
Essential Formulas:
Component Form: \( \vec{v} = \langle x_2 - x_1, y_2 - y_1 \rangle \)
Magnitude: \( |\vec{v}| = \sqrt{a^2 + b^2} \)
Direction: \( \theta = \tan^{-1}(b/a) \) (adjust for quadrant)
From Magnitude/Direction: \( \vec{v} = \langle r\cos\theta, r\sin\theta \rangle \)
Addition: \( \vec{u} + \vec{v} = \langle a_1 + a_2, b_1 + b_2 \rangle \)
Scalar Multiplication: \( k\vec{v} = \langle ka, kb \rangle \)
Unit Vector: \( \hat{v} = \frac{\vec{v}}{|\vec{v}|} \)
📚 Study Tips
✓ Always adjust direction angle based on which quadrant the vector is in
✓ Magnitude is always non-negative (like distance)
✓ Unit vectors have magnitude = 1
✓ Scalar multiplication changes magnitude, not direction (unless negative)
✓ Use head-to-tail method for adding vectors graphically