Vector Arithmetic

Given vectors \(\vec{v} = \langle -2, 2 \rangle\) and \(\vec{w} = \langle 5, 3 \rangle\), find \(\vec{v} + \vec{w}\). Graph the scenario.

Idea: Add vectors by adding the corresponding components.

\[ \begin{aligned} \vec{v} + \vec{w} &= \langle -2, 2 \rangle + \langle 5, 3 \rangle \\ &= \langle -2 + 5,\ 2 + 3 \rangle \\ &= \langle 3, 5 \rangle \end{aligned} \]

Notice the sum of the vectors is the vector corresponding to a diagonal of the parallelogram.

On a grid: v from the origin to (-2, 2), w from the origin to (5, 3), dashed lines completing the parallelogram, and v plus w from the origin to (3, 5) along the diagonal.

Given the vectors \(\vec{v} = \langle -2, 2 \rangle\) and \(\vec{w} = \langle 3, 5 \rangle\), find \(\vec{v} - \vec{w}\). Graph the scenario.

Idea: Subtract vectors by subtracting the corresponding components.

\[ \begin{aligned} \vec{v} - \vec{w} &= \langle -2, 2 \rangle - \langle 3, 5 \rangle \\ &= \langle -2 - 3,\ 2 - 5 \rangle \\ &= \langle -5, -3 \rangle \end{aligned} \]

Notice the difference of the vectors is the vector corresponding to the diagonal of the parallelogram connecting the terminal points. And it points from the terminal point of \(\vec{w}\) to the terminal point of \(\vec{v}\).

On a grid: v from the origin to (-2, 2), w from the origin to (3, 5), the parallelogram they form, a segment from the tip of w to the tip of v, and v minus w from the origin to (-5, -3).

Two vectors are parallel if they have the same direction, or the opposite direction. If \(k\) is a nonzero scalar, then \(k\vec{v}\) is parallel to \(\vec{v}\), and its magnitude is scaled by a factor of \(|k|\): \(\|k\vec{v}\| = |k|\,\|\vec{v}\|\).

Four arrows from the same starting point: v pointing right, one half v pointing right and half as long, 3v pointing right and three times as long, and negative 1.5 v pointing left and one and a half times as long.

Given the vector \(\vec{v} = \langle 1, 2 \rangle\), find \(2\vec{v}\). Graph the scenario.

Idea: Multiply the components by two.

\[ \begin{aligned} 2\vec{v} &= 2\langle 1, 2 \rangle \\ &= \langle 2 \cdot 1,\ 2 \cdot 2 \rangle \\ &= \langle 2, 4 \rangle \end{aligned} \]

Notice \(2\vec{v}\) is double the length of \(\vec{v}\) and they have the same direction.

On a grid: v from the origin to (1, 2), and 2v from the origin to (2, 4), in the same direction.

Given the vector \(\vec{v} = \langle 1, 2 \rangle\), find \(-2\vec{v}\). Graph the scenario.

\[ \begin{aligned} -2\vec{v} &= -2\langle 1, 2 \rangle \\ &= \langle -2 \cdot 1,\ -2 \cdot 2 \rangle \\ &= \langle -2, -4 \rangle \end{aligned} \]

Notice the direction of the vector is reversed!

On a grid: v from the origin to (1, 2), and negative 2v from the origin to (-2, -4), in the opposite direction.

For vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\), and scalars \(a\) and \(b\), the following are true.

\[ \vec{v} + \vec{w} = \vec{w} + \vec{v} \]
\[ (\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w}) \]
\[ \vec{v} + \vec{0} = \vec{v} \]
\[ \vec{v} - \vec{v} = \vec{0} \]
\[ 0\vec{v} = \vec{0} \]
\[ a(b\vec{v}) = (ab)\vec{v} \]
\[ a(\vec{v} + \vec{w}) = a\vec{v} + a\vec{w} \]
\[ (a + b)\vec{v} = a\vec{v} + b\vec{v} \]

Each of these properties can be verified by writing the vectors in components and carrying out the operations. As an example, we verify one of them below.

Verify the below property of vectors.

\[ a(\vec{v} + \vec{w}) = a\vec{v} + a\vec{w} \]

This follows by performing the operations via the components of the vectors, as below.

\[ \begin{aligned} \vec{v} &= \langle v_1, v_2, v_3 \rangle \\ \vec{w} &= \langle w_1, w_2, w_3 \rangle \end{aligned} \]
\[ \begin{aligned} &a\vec{v} + a\vec{w} \\ &= a\langle v_1, v_2, v_3 \rangle \\ &\qquad {}+ a\langle w_1, w_2, w_3 \rangle \\ &= \langle av_1, av_2, av_3 \rangle \\ &\qquad {}+ \langle aw_1, aw_2, aw_3 \rangle \\ &= \langle av_1 + aw_1,\ av_2 + aw_2,\ av_3 + aw_3 \rangle \\ &= a\langle v_1 + w_1,\ v_2 + w_2,\ v_3 + w_3 \rangle \\ &= a(\vec{v} + \vec{w}) \end{aligned} \]

The other properties can be proven in a similar manner.