Vectors and Their Applications in Calculus

Vectors in Calculus

Introduction to Vectors

Vectors are mathematical objects that have both magnitude and direction. They are fundamental in calculus, especially in multivariable calculus and physics applications. Vectors are often represented in component form using unit vectors i, j, and k for the x, y, and z axes, respectively.

• Vector Notation: A vector v in three dimensions can be written as v = a i + b j + c k, where a, b, and c are real numbers.

• Magnitude of a Vector: The length of vector v = a i + b j + c k is $|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}$.

• Unit Vector: A vector of length 1 in the direction of v is $\mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|}$.

Vector Operations

Dot Product

The dot product (or scalar product) of two vectors produces a scalar and is useful for finding angles and projections.

• Definition: For vectors $\mathbf{v} = a_1 i + b_1 j + c_1 k$ and $\mathbf{w} = a_2 i + b_2 j + c_2 k$, the dot product is $\mathbf{v} \cdot \mathbf{w} = a_1 a_2 + b_1 b_2 + c_1 c_2$.

• Geometric Interpretation: $\mathbf{v} \cdot \mathbf{w} = |\mathbf{v}| |\mathbf{w}| \cos \theta$, where $\theta$ is the angle between the vectors.

• Example: If $\mathbf{v} = 3i + 2j + k$ and $\mathbf{w} = 5i + 6j + 2k$, then $\mathbf{v} \cdot \mathbf{w} = 3 \times 5 + 2 \times 6 + 1 \times 2 = 15 + 12 + 2 = 29$.

Cross Product

The cross product of two vectors in three dimensions results in a vector that is orthogonal to both original vectors.

• Definition: $\mathbf{v} \times \mathbf{w} = (b_1 c_2 - c_1 b_2)i - (a_1 c_2 - c_1 a_2)j + (a_1 b_2 - b_1 a_2)k$

• Magnitude: $|\mathbf{v} \times \mathbf{w}| = |\mathbf{v}| |\mathbf{w}| \sin \theta$

• Direction: Determined by the right-hand rule.

• Example: $\mathbf{v} = 3i - 2j - 4k$, $\mathbf{w} = -2i + 4j + 3k$ $\mathbf{v} \times (4\mathbf{w}) = 40i - 4j + 32k$

Scalar and Vector Projections

Projections are used to find the component of one vector along the direction of another.

• Scalar Projection (comp): $\mathrm{comp}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}$

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

• Example: $\mathbf{v} = i + j + k$, $\mathbf{u} = 12i + 3j + 4k$ $\mathrm{proj}_{\mathbf{v}} \mathbf{u} = \frac{19}{3}i + \frac{19}{3}j + \frac{19}{3}k$

Orthogonality of Vectors

Two vectors are orthogonal (perpendicular) if their dot product is zero.

• Test for Orthogonality: $\mathbf{v} \cdot \mathbf{w} = 0$

• Example: Which vector is orthogonal to $20i - 8j$? $-10i - 25j$ is orthogonal because $20 \times (-10) + (-8) \times (-25) = -200 + 200 = 0$.

Applications of Vectors

Work Done by a Force

Work is the dot product of force and displacement vectors. If the force is applied at an angle, only the component in the direction of motion does work.

• Formula: $W = \mathbf{F} \cdot \mathbf{d} = |\mathbf{F}| |\mathbf{d}| \cos \theta$

• Example: Pulling a box 6 meters with a 158 N force at 15°: $W = 158 \times 6 \times \cos(15^\circ) = 915.7$ Joules

Components of Velocity

When an object is projected at an angle, its velocity can be resolved into horizontal and vertical components using trigonometry.

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

• Example: Bullet fired at 1163 ft/sec at 31° above horizontal: $v_y = 1163 \times \sin(31^\circ) = 599.0$ ft/sec

Area of a Parallelogram

The area of a parallelogram defined by two vectors is the magnitude of their cross product.

• Formula: $A = |\mathbf{u} \times \mathbf{v}|$

• Example: Given points $P, Q, R, S$, find vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$, then $A = |\overrightarrow{PQ} \times \overrightarrow{PR}| = \sqrt{14829}$

Torque

Torque is a measure of the tendency of a force to rotate an object about an axis, fulcrum, or pivot.

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

• Example: $|\overrightarrow{PQ}| = 9$ in, $|\mathbf{F}| = 5$ lb, $\theta = 45^\circ$ $\tau = 9 \times 5 \times \sin(45^\circ) = \frac{15}{8} \sqrt{2}$ ft-lb

Triple Scalar Product

The triple scalar product of three vectors gives the volume of the parallelepiped they define.

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

• Example: $\mathbf{u} = i + j + k$, $\mathbf{v} = 8i + 5j + 2k$, $\mathbf{w} = 9i - 6j + 4k$ $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = -3$

Summary Table: Key Vector Operations

Operation	Formula	Result	Example
Dot Product	$\mathbf{v} \cdot \mathbf{w}$	Scalar	$3i + 2j + k$ and $5i + 6j + 2k$ $\rightarrow 29$
Cross Product	$\mathbf{v} \times \mathbf{w}$	Vector	$3i - 2j - 4k$ and $-2i + 4j + 3k$ $\rightarrow 40i - 4j + 32k$
Scalar Projection	$\frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|}$	Scalar	$u = 4i + 5j$, $v = i + j$ $\rightarrow \frac{9\sqrt{2}}{2}$
Vector Projection	$\left( \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \right) \mathbf{v}$	Vector	$u = 12i + 3j + 4k$, $v = i + j + k$ $\rightarrow \frac{19}{3}i + \frac{19}{3}j + \frac{19}{3}k$
Work	$|\mathbf{F}| |\mathbf{d}| \cos \theta$	Scalar	$F = 158$ N, $d = 6$ m, $\theta = 15^\circ$ $\rightarrow 915.7$ J
Torque	$|\mathbf{r}| |\mathbf{F}| \sin \theta$	Scalar	$r = 9$ in, $F = 5$ lb, $\theta = 45^\circ$ $\rightarrow \frac{15}{8} \sqrt{2}$ ft-lb

Additional info:

• These problems are typical of a Calculus II or Multivariable Calculus course, focusing on vector algebra and its applications in physics and geometry.

• Understanding vector operations is essential for later topics such as line integrals, surface integrals, and applications in engineering and physics.