Vectors, Lines, and Planes in Multivariable Calculus

Vectors: Preview and Basic Operations

Vector Definition and Notation

Vectors are mathematical objects that have both magnitude and direction. In calculus, vectors are often used to represent points, directions, and quantities in space.

• Point notation: A point P is denoted as P(x1, y1), and Q as Q(x2, y2).

• Vector from P to Q: The vector \( \vec{PQ} \) is given by: $\vec{PQ} = \langle x_2 - x_1, y_2 - y_1 \rangle$

• Direction: The vector points from P to Q, with its components representing the change in x and y.

• Example: If P(1,2) and Q(3,5), then $\vec{PQ} = \langle 3-1, 5-2 \rangle = \langle 2, 3 \rangle$

• Reverse direction: $\vec{QP} = -\vec{PQ}$

Magnitude (Length) of a Vector

The magnitude or length of a vector \( \vec{u} = \langle x, y \rangle \) is calculated using the Pythagorean theorem.

• Formula: $|\vec{u}| = \sqrt{x^2 + y^2}$

• Example: For \( \vec{u} = \langle 1, 2 \rangle \), $|\vec{u}| = \sqrt{1^2 + 2^2} = \sqrt{5}$

• The same formula applies in higher dimensions, adding more squared components.

Vector Addition and Subtraction

Vectors can be added or subtracted component-wise.

• Addition: $\vec{v} + \vec{w} = \langle v_1 + w_1, v_2 + w_2, v_3 + w_3 \rangle$

• Subtraction: $\vec{v} - \vec{w} = \langle v_1 - w_1, v_2 - w_2, v_3 - w_3 \rangle$

• Example: $\vec{v} = \langle 1, 2, 3 \rangle$, $\vec{w} = \langle 4, 5, 6 \rangle$ $\vec{v} + \vec{w} = \langle 5, 7, 9 \rangle$ $\vec{v} - \vec{w} = \langle -3, -3, -3 \rangle$

Unit Vectors

A unit vector is a vector with magnitude 1, often used to indicate direction.

• Definition: A unit vector in the direction of \( \vec{v} \) is $\frac{\vec{v}}{|\vec{v}|}$

• Example: For \( \vec{v} = \langle 1, 2, 3 \rangle \), $|\vec{v}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14}$ Unit vector: $\frac{\vec{v}}{|\vec{v}|} = \langle \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \rangle$

• Opposite direction: $-\frac{\vec{v}}{|\vec{v}|}$

• Scaling: A vector with length 3 in the same direction as \( \vec{v} \) is $3 \frac{\vec{v}}{|\vec{v}|}$

Shapes in Higher Dimensions

Planes and Spheres in 3D

Many geometric shapes in three dimensions (\( \mathbb{R}^3 \)) are extensions of their two-dimensional counterparts.

• Plane: The equation $x = 5$ in \( \mathbb{R}^2 \) represents a vertical line; in \( \mathbb{R}^3 \), it represents a plane where x is always 5 and y, z can be any values.

• Sphere: The equation $(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2$ describes a sphere with center (h, k, l) and radius r.

Vector Cross Product

Definition and Calculation

The cross product of two vectors in \( \mathbb{R}^3 \) produces a vector perpendicular to both original vectors.

• Formula: For \( \vec{u} = \langle u_1, u_2, u_3 \rangle \), \( \vec{v} = \langle v_1, v_2, v_3 \rangle \): $\vec{u} \times \vec{v} = \langle u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 \rangle$

• Properties: $\vec{v} \times \vec{u} = - (\vec{u} \times \vec{v})$

• Example: $\vec{u} = \langle 2, 1, 1 \rangle$, $\vec{v} = \langle 5, 0, 1 \rangle$ $\vec{u} \times \vec{v} = \langle 1, 7, -5 \rangle$

Lines and Planes in Space

Vector and Parametric Forms of a Line

A line in space is defined by a point and a direction vector. The vector form and parametric form are used to describe lines in \( \mathbb{R}^3 \).

• Vector form: $\vec{P}(t) = \vec{r}_0 + t \vec{v}$ where \( \vec{r}_0 \) is the position vector of a point on the line, \( \vec{v} \) is the direction vector, and t is a real parameter.

• Parametric form: $x(t) = x_0 + a t$ $y(t) = y_0 + b t$ $z(t) = z_0 + c t$ where \( \vec{v} = \langle a, b, c \rangle \).

Example: Line Through Two Points

To find the equation of a line through points P and Q:

• Direction vector: $\vec{PQ} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle$

• Position vector: $\vec{r}_0$ from origin to P

• Vector form: $\vec{P}(t) = \vec{r}_0 + t \vec{PQ}$

• Parametric form: $x = x_0 + t(x_2 - x_1)$, $y = y_0 + t(y_2 - y_1)$, $z = z_0 + t(z_2 - z_1)$

Line Segment Between Two Points

To describe the segment from P to Q, restrict t to [0,1].

• At t=0, the point is at P; at t=1, the point is at Q.

• Segment: $\vec{P}(t) = \vec{r}_0 + t \vec{PQ}$, $0 \leq t \leq 1$

Planes in Space

A plane in \( \mathbb{R}^3 \) can be defined by a point and a normal vector.

• Equation: $a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$ where \( \vec{n} = \langle a, b, c \rangle \) is the normal vector, and (x0, y0, z0) is a point on the plane.

• The normal vector is perpendicular to the plane.

Example: Plane Through Vectors and a Point

To find the equation of a plane containing two vectors and a point:

• Find the normal vector by taking the cross product of the two vectors.

• Plug the normal vector and the point into the plane equation.

• Example: If \( \vec{u} = \langle 0, 1, 2 \rangle \), \( \vec{v} = \langle -1, -3, 0 \rangle \), and point (-4, 7, 5): $\vec{n} = \vec{u} \times \vec{v} = \langle 6, -2, 1 \rangle$ Plane equation: $6(x+4) - 2(y-7) + (z-5) = 0$

Relationships Between Lines and Planes

Skew Lines

In \( \mathbb{R}^2 \), lines with the same slope must intersect. In \( \mathbb{R}^3 \), lines can be skew—never intersecting and not parallel.

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

• Example: $\vec{u}(t) = \langle 1, 0, 1 \rangle + t \langle 1, 2, 3 \rangle$ and $\vec{v}(t) = t \langle 1, 0, 0 \rangle$ are skew.

Parallel and Orthogonal Planes

Planes are parallel if their normal vectors are parallel, and orthogonal if their normal vectors are perpendicular.

• Parallel planes: $\vec{n}_1 \parallel \vec{n}_2$

• Orthogonal planes: $\vec{n}_1 \cdot \vec{n}_2 = 0$

• Intersection: The intersection of two planes is a line. The direction vector of the intersection line is the cross product of the normal vectors.

Supplemental Example: Intersection of Lines

To determine if two objects traveling along lines will intersect, solve for parameters t and s such that their position vectors are equal.

• Example: $\vec{P}_1(s) = \langle 2+3s, 4+2s, 3+5s \rangle$ $\vec{P}_2(t) = \langle 5+2t, 3+5t, -5+5t \rangle$

• Set up equations for each coordinate and solve for t and s.

• If a solution exists, the objects intersect at those values.

Additional info: These notes cover foundational concepts in multivariable calculus, including vectors, vector operations, lines, planes, and their relationships in three-dimensional space. The material is directly relevant to Calculus topics, especially those found in chapters on vectors, lines, and planes.