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Vector Operations

Vector Operations

A vector is a mathematical quantity that has both magnitude (length) and direction. In a 2D coordinate plane, vectors are commonly written in component form as v⃗=⟨a,b⟩\vec{v} = \langle a, b \rangle, where aa represents the horizontal shift and bb represents the vertical shift.

Basic Vector Operations

When working with vectors in component form, operations are performed component by component.

  • Addition: To add two vectors, add their corresponding components. ⟨u1,u2⟩+⟨v1,v2⟩=⟨u1+v1,u2+v2⟩\langle u_1, u_2 \rangle + \langle v_1, v_2 \rangle = \langle u_1 + v_1, u_2 + v_2 \rangle
  • Subtraction: To subtract two vectors, subtract their corresponding components. ⟨u1,u2⟩−⟨v1,v2⟩=⟨u1−v1,u2−v2⟩\langle u_1, u_2 \rangle - \langle v_1, v_2 \rangle = \langle u_1 - v_1, u_2 - v_2 \rangle
  • Scalar Multiplication: To multiply a vector by a real number (a scalar) kk, multiply each component by kk. This scales the vector's length and can reverse its direction if kk is negative. k⟨a,b⟩=⟨ka,kb⟩k\langle a, b \rangle = \langle ka, kb \rangle

Magnitude of a Vector

The magnitude (or length) of a vector v⃗=⟨a,b⟩\vec{v} = \langle a, b \rangle is denoted by ∣v⃗∣|\vec{v}| or ∣∣v⃗∣∣||\vec{v}||. It can be found using the Pythagorean theorem:

∣v⃗∣=a2+b2|\vec{v}| = \sqrt{a^2 + b^2}

Unit Vectors

A unit vector is a vector that has a magnitude of exactly 11. To find a unit vector u^\hat{u} that points in the exact same direction as a given vector v⃗\vec{v}, you divide the vector by its own magnitude:

u^=1∣v⃗∣v⃗=⟨a∣v⃗∣,b∣v⃗∣⟩\hat{u} = \frac{1}{|\vec{v}|} \vec{v} = \left\langle \frac{a}{|\vec{v}|}, \frac{b}{|\vec{v}|} \right\rangle

Example Problems

Example 1: Given u⃗=⟨3,4⟩\vec{u} = \langle 3, 4 \rangle and v⃗=⟨−1,2⟩\vec{v} = \langle -1, 2 \rangle, find 2u⃗−3v⃗2\vec{u} - 3\vec{v} and ∣2u⃗−3v⃗∣|2\vec{u} - 3\vec{v}|.

  1. First, find scalar multiples: 2u⃗=2⟨3,4⟩=⟨6,8⟩2\vec{u} = 2\langle 3, 4 \rangle = \langle 6, 8 \rangle 3v⃗=3⟨−1,2⟩=⟨−3,6⟩3\vec{v} = 3\langle -1, 2 \rangle = \langle -3, 6 \rangle
  2. Next, subtract the scaled vectors: 2u⃗−3v⃗=⟨6−(−3),8−6⟩=⟨9,2⟩2\vec{u} - 3\vec{v} = \langle 6 - (-3), 8 - 6 \rangle = \langle 9, 2 \rangle
  3. Finally, find the magnitude of the resulting vector: ∣2u⃗−3v⃗∣=92+22=81+4=85|2\vec{u} - 3\vec{v}| = \sqrt{9^2 + 2^2} = \sqrt{81 + 4} = \sqrt{85}

Example 2: Find the unit vector in the direction of ⟨−5,12⟩\langle -5, 12 \rangle.

  1. Let v⃗=⟨−5,12⟩\vec{v} = \langle -5, 12 \rangle. First, calculate its magnitude: ∣v⃗∣=(−5)2+122=25+144=169=13|\vec{v}| = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
  2. Divide the vector by its magnitude to get the unit vector: v^=113⟨−5,12⟩=⟨−513,1213⟩\hat{v} = \frac{1}{13} \langle -5, 12 \rangle = \left\langle -\frac{5}{13}, \frac{12}{13} \right\rangle