OddClasses — Odd ones figure it out.

Vectors

A vector has both magnitude and direction. In

\mathbb{R}^2

we write

\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}

(x,y)

. The zero vector

\mathbf{0}

has magnitude 0 and no direction.

Theorem 1📌 Learn for exam

Vector Operations

For vectors $\mathbf{a} = (a_1, a_2)$ and $\mathbf{b} = (b_1, b_2)$ , scalar $k$ :

\begin{aligned} \mathbf{a} + \mathbf{b} &= (a_1+b_1,\; a_2+b_2) \\ k\mathbf{a} &= (ka_1,\; ka_2) \\ \|\mathbf{a}\| &= \sqrt{a_1^2 + a_2^2} \end{aligned}

▸ Vector Addition — drag the tips of a and bboth tips are draggable

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Theorem 2📌 Learn for exam

Dot Product

\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta

Key facts: $\mathbf{a} \cdot \mathbf{b} = 0 \iff \mathbf{a} \perp \mathbf{b}$ . The dot product is a scalar, not a vector.

▸ Dot Product — drag vectors to change the anglewatch the projection (teal dot)

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Tip

When

\mathbf{a} \cdot \mathbf{b} > 0

the angle between them is acute. When

< 0

it is obtuse. When

= 0

they are perpendicular.

Worked Example

Given $\mathbf{a} = (3, -1)$ and $\mathbf{b} = (2, 4)$ , find $\mathbf{a}\cdot\mathbf{b}$ and the angle between them.