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13.1 - 13.4 Review of Vectors

\( P(x_1, y_1), \quad Q(x_2, y_2) \)

vector from \( P \) to \( Q \):

\[ \vec{PQ} = \langle x_2 - x_1, y_2 - y_1 \rangle \]\[ = (x_2 - x_1)\vec{i} + (y_2 - y_1)\vec{j} \]
Coordinate plane showing points P and Q with their respective x and y coordinates labeled on the axes.

destination \( x \) or \( y \) minus starting \( x \) or \( y \)

for example,

\( P(1, 2) \quad Q(3, -5) \)

\[ \vec{PQ} = \langle 3-1, -5-2 \rangle = \langle 2, -7 \rangle \]
  • 2 "steps" right
  • 7 "steps" down

notice \( \vec{QP} = -\vec{PQ} \)

\[ \vec{QP} = \langle 1-3, 2 - (-5) \rangle = \langle -2, 7 \rangle \]

minus reverses direction

Diagram showing two opposite vectors between points P and Q: vector PQ in red and vector QP in green.

higher dimensions, same idea

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magnitude / length of vector:

Vector u in a 2D coordinate system with components 1 and 2, forming a right triangle.

\( \vec{u} = \langle 1, 2 \rangle \)

its length is

\[ |\vec{u}| = \sqrt{1^2 + 2^2} = \sqrt{5} \]

same idea in higher dimensions

Addition / Subtraction:

\( \vec{v} = \langle 1, 2, 3 \rangle \)

\( \vec{w} = \langle 4, 5, 6 \rangle \)

\[ \vec{v} + \vec{w} = \langle 1+4, 2+5, 3+6 \rangle = \langle 5, 7, 9 \rangle \]\[ \vec{v} - \vec{w} = \langle 1-4, 2-5, 3-6 \rangle = \langle -3, -3, -3 \rangle \]
\[ \vec{v} - \vec{w} = \vec{v} + (-\vec{w}) \]
Vector addition diagram showing vector v and vector w placed tip-to-tail to result in vector v+w.
Vector subtraction diagram showing vector v plus negative vector w resulting in vector v-w.
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Unit Vectors

Unit vector: a vector with magnitude of 1

\[ \vec{v} = \langle 1, 2, 3 \rangle \text{ is NOT a unit vector} \]

because

\[ |\vec{v}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14} \neq 1 \]

Finding Vectors in Specific Directions

A unit vector in the same direction as \( \vec{v} \):

\[ \frac{\vec{v}}{|\vec{v}|} = \frac{\langle 1, 2, 3 \rangle}{\sqrt{14}} = \left\langle \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \right\rangle \]

Opposite direction as \( \vec{v} \):

\[ -\frac{\vec{v}}{|\vec{v}|} \]

Vector with length 3 in the same direction as \( \vec{v} \):

\[ 3 \frac{\vec{v}}{|\vec{v}|} \]
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Shapes in 3D Space

Most shapes in 3D (\( \mathbb{R}^3 \)) are very similar to their counterparts in 2D (\( \mathbb{R}^2 \)).

Example in 2D

For example, \( x = 5 \) in \( \mathbb{R}^2 \rightarrow \) all points with \( x \) of 5 and all possible \( y \).

A 2D Cartesian coordinate system showing a vertical dashed line at x=5 parallel to the y-axis.

Example in 3D

In 3D, \( x = 5 \) is a collection of all points with \( x = 5 \), and \( y, z \) all reals. This is a plane.

A 3D coordinate system with x, y, and z axes showing a vertical plane intersecting the x-axis at 5.

Equation of a Sphere

Sphere:

\[ (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 \]
  • Center: \( (h, k, l) \)
  • Radius: \( r \)
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vector dot product:

\(\vec{u} = \langle 1, 2, 3 \rangle\)

\(\vec{v} = \langle 4, 5, 6 \rangle\)

\(\vec{u} \cdot \vec{v} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 32\) a scalar

notice \(\vec{v} \cdot \vec{u} = \vec{u} \cdot \vec{v}\)

another formula:

\(\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos \theta\)

Diagram showing two vectors u and v originating from the same point with an angle theta between them.

notice if \(\vec{u} \cdot \vec{v} = 0\), then \(\vec{u} \perp \vec{v}\)

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vector cross product

\(\vec{u} = \langle 2, 1, 1 \rangle\)     \(\vec{v} = \langle 5, 0, 1 \rangle\)

\[ \vec{u} \times \vec{v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 2 & 1 & 1 \\ 5 & 0 & 1 \end{vmatrix} \]

\[ = \vec{i} \begin{vmatrix} 1 & 1 \\ 0 & 1 \end{vmatrix} - \vec{j} \begin{vmatrix} 2 & 1 \\ 5 & 1 \end{vmatrix} + \vec{k} \begin{vmatrix} 2 & 1 \\ 5 & 0 \end{vmatrix} \]

\[ = \vec{i} (1 \cdot 1 - 0 \cdot 1) - \vec{j} (2 - 5) + \vec{k} (0 - 5) \]

\[ = \vec{i} + 3\vec{j} - 5\vec{k} = \langle 1, 3, -5 \rangle \]

\(\vec{v} \times \vec{u} = - (\vec{u} \times \vec{v})\)