Lecture 1: Multivariable Calculus

Vector: arrow pointing in the direction of a displacement

Displacement: length of the arrow (magnitude of the vector)

$$\begin{bmatrix}v_1 \\v_2\end{bmatrix}$$ + $$\begin{bmatrix}w_1 \\w_2\end{bmatrix}$$ =$$\begin{bmatrix}v_1 & w_1\\v_2 & w_2\end{bmatrix}$$

Subtracting Vectors: $$\hat{v} - \hat{w} = \hat{v} + (-\hat{w})$$

Multiplying Vectors by a Scalar:

$$a \begin{bmatrix}x_1\\x_2\end{bmatrix}= \begin{bmatrix}a x_1 \\ a x_2 \end{bmatrix}$$

$$3 \hat{v}$$ increases the magnitude of $$\hat{v}$$ three magnitudes.

$$-\hat{v}$$ reverses the direction of $$\hat{v}$$

Position Vectors -Vector with an initial point at it’s origin and terminal point at $$P(v_1, v_2)$$ -Any vector can be translated into a position vector Suppose we have two points: $$P_1(x_1, y_1)$$ and $$P_2(x_2, y_2)$$ $$\hat{v} = \vec{P_1 P_2} = <x_2 - x_1, y_2 - y_1>$$

Calculating Magnitude $$\lVert v\rVert = \sqrt{(v_1)^2 + (v_2)^2}$$

Unit Vectors A vector with length/magnitude of 1. Take a vector divided by magnitude. $$\hat{u} = \frac{\hat{v}}{\lVert u\rVert}$$

Standard Basis Vectors $$\hat{i} = <1,0>$$ $$\hat{j} = <0,1>$$

In $$R^2$$ there are two standard basis vectors. In $$R^3$$ there are three standard basis vectors.

What is a subspace of $$R_2$$? -A line through the origin is a subspace of dimension 1 -A plane through the origin is a subspace of dimension 2