Vectors and Applications in Calculus: Study Guide

Vectors in Calculus

Vector Operations

Vectors are fundamental objects in calculus and physics, representing quantities with both magnitude and direction. Common operations include addition, scalar multiplication, dot product, and cross product.

• Vector Addition: Combine corresponding components of two vectors.

• Scalar Multiplication: Multiply each component of a vector by a scalar.

• Dot Product: For vectors $\mathbf{v} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ and $\mathbf{w} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}$, the dot product is: $\mathbf{v} \cdot \mathbf{w} = a_1b_1 + a_2b_2 + a_3b_3$

• Cross Product: Produces a vector orthogonal to both input vectors. $\mathbf{v} \times \mathbf{w} = (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}$

Example:

Find the dot product of $\mathbf{v} = 3\mathbf{i} + 2\mathbf{j} + \mathbf{k}$ and $\mathbf{w} = 5\mathbf{i} + 6\mathbf{j} + 2\mathbf{k}$:

$\mathbf{v} \cdot \mathbf{w} = 3 \times 5 + 2 \times 6 + 1 \times 2 = 15 + 12 + 2 = 29$

Angle Between Vectors

The angle $\theta$ between two vectors $\mathbf{v}$ and $\mathbf{w}$ can be found using the dot product:

• $\cos \theta = \dfrac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|}$

• To find $\theta$, use $\theta = \arccos\left(\dfrac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|}\right)$

Example:

For $\mathbf{v} = \mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{w} = \mathbf{i} - \mathbf{j} - \mathbf{k}$, the angle is approximately $35.3^\circ$.

Vector Projections and Scalar Projections

Projection is the component of one vector along the direction of another.

• Vector Projection: $\mathrm{proj}_{\mathbf{v}} \mathbf{u} = \left( \dfrac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \right) \mathbf{v}$

• Scalar Projection: $\mathrm{scal}_{\mathbf{v}} \mathbf{u} = \dfrac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|}$

Example:

Find $\mathrm{scal}_{\mathbf{v}} \mathbf{u}$ for $\mathbf{u} = 4\mathbf{i} + 5\mathbf{j}$, $\mathbf{v} = \mathbf{i} + \mathbf{j}$:

$\mathrm{scal}_{\mathbf{v}} \mathbf{u} = \dfrac{4 \times 1 + 5 \times 1}{\sqrt{1^2 + 1^2}} = \dfrac{9}{\sqrt{2}} = \dfrac{9\sqrt{2}}{2}$

Orthogonality of Vectors

Two vectors are orthogonal if their dot product is zero.

• Given $\mathbf{a} = 20\mathbf{i} - 8\mathbf{j}$, a vector $\mathbf{b}$ is orthogonal to $\mathbf{a}$ if $\mathbf{a} \cdot \mathbf{b} = 0$.

Example:

$\mathbf{b} = -10\mathbf{i} - 25\mathbf{j}$ is orthogonal to $\mathbf{a}$.

Applications of Vectors

Velocity Components

Velocity can be decomposed into horizontal and vertical components using trigonometry.

• Vertical Component: $v_y = v \sin \theta$

• Horizontal Component: $v_x = v \cos \theta$

Example:

A bullet fired at $1163$ft/sec at $31^\circ$ above the horizontal has a vertical component:

$v_y = 1163 \sin 31^\circ \approx 599.0$ ft/sec

Work Done by a Force

Work is the product of the force component in the direction of motion and the displacement.

• $W = Fd \cos \theta$

Example:

Sliding a box $6$meters with a $158$ N force at $15^\circ$:

$W = 158 \times 6 \times \cos 15^\circ \approx 915.7$ Joules

Triple Scalar Product

The triple scalar product $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$ gives the volume of the parallelepiped formed by three vectors.

• $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$

Example:

Given $\mathbf{u} = \mathbf{i} + \mathbf{j} + \mathbf{k}$, $\mathbf{v} = 8\mathbf{i} + 5\mathbf{j} + 2\mathbf{k}$, $\mathbf{w} = 9\mathbf{i} + 6\mathbf{j} + 4\mathbf{k}$, the triple scalar product is $-3$.

Area of a Parallelogram Determined by Points

The area of a parallelogram defined by vectors $\mathbf{a}$ and $\mathbf{b}$ is $\|\mathbf{a} \times \mathbf{b}\|$.

Example:

Given points $P, Q, R, S$, the area is $\sqrt{14829}$.

Torque

Torque is a measure of the rotational force applied at a point, calculated as the cross product of the position vector and the force vector.

• $\tau = \|\mathbf{r}\| \|\mathbf{F}\| \sin \theta$

Example:

For $|\overrightarrow{PQ}| = 9$ in, $|\overrightarrow{F}| = 5$ lb, $\theta = 45^\circ$:

$\tau = 9 \times 5 \times \sin 45^\circ = \dfrac{15}{8} \sqrt{2}$ ft-lb

Summary Table: Key Vector Operations

Operation	Formula	Application
Dot Product	$\mathbf{v} \cdot \mathbf{w}$	Angle, work, orthogonality
Cross Product	$\mathbf{v} \times \mathbf{w}$	Area, orthogonal vector, torque
Projection	$\mathrm{proj}_{\mathbf{v}} \mathbf{u}$	Component along direction
Triple Scalar Product	$\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$	Volume of parallelepiped
Work	$W = Fd \cos \theta$	Physics, energy
Torque	$\tau = \|\mathbf{r}\| \|\mathbf{F}\| \sin \theta$	Rotational force

Conclusion

Understanding vector operations and their applications is essential in calculus, especially for solving problems in physics and engineering. Mastery of these concepts enables students to analyze forces, motion, and geometric properties in multidimensional spaces.