# Lecture 1: Multivariable Calculus

**Vector:** arrow pointing in the direction of a displacement

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

**Adding Vectors:**

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

**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 itâ€™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