Vectors and Lines in Multivariable Calculus: Study Notes

Vectors in Three Dimensions

Distance Between Points

In three-dimensional space, the distance between two points P and Q with coordinates $P = (x_1, y_1, z_1)$ and $Q = (x_2, y_2, z_2)$ is given by the distance formula:

• Formula:

$\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

• Example: For $P = (-1, 1, 1)$ and $Q = (0, 3, -1)$:

$\text{Distance} = \sqrt{(0 - (-1))^2 + (3 - 1)^2 + ((-1) - 1)^2} = \sqrt{1^2 + 2^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3$

• Application: The set of all points at a fixed distance $r$ from a point $P$ forms a sphere of radius $r$ centered at $P$:

$(x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 = r^2$

Vectors and Their Properties

A vector in three dimensions is an ordered triple $\vec{v} = \langle a, b, c \rangle$ representing magnitude and direction.

• Displacement Vector: The vector from $P$ to $Q$ is $\vec{v} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle$.

• Magnitude (Length): $|\vec{v}| = \sqrt{a^2 + b^2 + c^2}$

• Unit Vector: $\vec{u} = \frac{\vec{v}}{|\vec{v}|}$ has length 1 and points in the same direction as $\vec{v}$.

Example: For $\vec{v} = \langle 1, 2, -2 \rangle$, $|\vec{v}| = 3$, so $\vec{u} = \langle \frac{1}{3}, \frac{2}{3}, -\frac{2}{3} \rangle$.

Dot Product and Angle Between Vectors

Dot Product

The dot product of two vectors $\vec{a}$ and $\vec{b}$ is:

$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$

• Geometric Interpretation: $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

• Orthogonality: Vectors are orthogonal (perpendicular) if $\vec{a} \cdot \vec{b} = 0$.

Example: For $\vec{a} = \langle 1, -2, 3 \rangle$, $\vec{b} = \langle 2, 1, 1 \rangle$:

$\vec{a} \cdot \vec{b} = (1)(2) + (-2)(1) + (3)(1) = 2 - 2 + 3 = 3$

$|\vec{a}| = \sqrt{1^2 + (-2)^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$ $|\vec{b}| = \sqrt{2^2 + 1^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$ $\cos \theta = \frac{3}{\sqrt{14}\sqrt{6}}$

Projection of Vectors

The scalar projection of $\vec{b}$ onto $\vec{a}$ is:

$\text{Scalar}_{\vec{a}} \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}$

The vector projection of $\vec{b}$ onto $\vec{a}$ is:

$\text{Proj}_{\vec{a}} \vec{b} = \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2} \right) \vec{a}$

Cross Product and Orthogonal Vectors

Cross Product

The cross product $\vec{a} \times \vec{b}$ produces a vector orthogonal to both $\vec{a}$ and $\vec{b}$:

$\vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

Example: For $\vec{a} = \langle 1, -2, 3 \rangle$, $\vec{b} = \langle 2, 1, 1 \rangle$:

$\vec{a} \times \vec{b} = \langle (-2)(1) - (3)(1), (3)(2) - (1)(1), (1)(1) - (-2)(2) \rangle = \langle -2 - 3, 6 - 1, 1 + 4 \rangle = \langle -5, 5, 5 \rangle$

• Magnitude: $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta$

• Area: The magnitude of the cross product gives the area of the parallelogram formed by $\vec{a}$ and $\vec{b}$; half of this is the area of the triangle.

Equations of Lines in Space

Parametric and Vector Equations

A line in space can be described using parametric equations or a vector equation.

• Parametric Equations:

$x = a t + x_0$ $y = b t + y_0$ $z = c t + z_0$

• Vector Equation:

$\vec{r}(t) = \langle x_0, y_0, z_0 \rangle + t \langle a, b, c \rangle$

• Direction Vector: $\langle a, b, c \rangle$ gives the direction of the line.

• Point on Line: $\langle x_0, y_0, z_0 \rangle$ is a point through which the line passes.

Example: Line through $(1, 2, 3)$ and $(0, 4, 4)$:

Direction vector: $\langle 0 - 1, 4 - 2, 4 - 3 \rangle = \langle -1, 2, 1 \rangle$ $\vec{r}(t) = \langle 1, 2, 3 \rangle + t \langle -1, 2, 1 \rangle$ Parametric: $x = 1 - t$, $y = 2 + 2t$, $z = 3 + t$

Intersection and Parallelism of Lines

To determine if two lines in space intersect, are parallel, or are skew:

• Intersection: Set parametric equations equal and solve for $t$ and $s$.

• Parallel: Direction vectors are scalar multiples.

• Skew: Lines are not parallel and do not intersect.

Example Table:

Condition	Test	Result
Intersection	Set $x_1(t) = x_2(s)$, $y_1(t) = y_2(s)$, $z_1(t) = z_2(s)$	Solution for $t$, $s$ exists
Parallel	Direction vectors proportional	Lines never meet
Skew	No intersection, not parallel	Lines are skew

Lines: Function and Parametric Forms

Single Variable (Function Form)

The equation of a line in the plane is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Point-Slope Form: $y - y_0 = m(x - x_0)$

Parametric Form

For a line in the plane:

$x = a t + x_0$ $y = b t + y_0$

• Slope: $\frac{b}{a}$

• Direction: Parallel to $\langle a, b \rangle$

Summary Table: Vector Operations

Operation	Formula	Result
Dot Product	$\vec{a} \cdot \vec{b}$	Scalar
Cross Product	$\vec{a} \times \vec{b}$	Vector orthogonal to both
Magnitude	$|\vec{v}|$	Length of vector
Unit Vector	$\vec{u} = \frac{\vec{v}}{|\vec{v}|}$	Vector of length 1
Projection	$\text{Proj}_{\vec{a}} \vec{b}$	Component of $\vec{b}$ in direction of $\vec{a}$

Key Concepts and Applications

• Distance formula is used to find the length between two points in space.

• Dot product determines angle and orthogonality between vectors.

• Cross product finds a vector perpendicular to two given vectors and computes area.

• Parametric equations describe lines in space and are useful for intersection and parallelism tests.

• Unit vectors are used to indicate direction only, with magnitude 1.