Dot Product and Vector Projections in Calculus

Vectors and Their Products

Types of Vector Products

In calculus and linear algebra, vectors can be combined using different products. The two main types are the dot product and the cross product.

• Dot Product: Produces a scalar from two vectors.

• Cross Product: Produces a vector perpendicular to the two input vectors (not covered in detail here).

Dot Product of Vectors

Definition and Calculation

The dot product of two vectors is a scalar quantity that measures the extent to which two vectors point in the same direction.

• If $\vec{v} = \langle x_1, y_1, z_1 \rangle$ and $\vec{w} = \langle x_2, y_2, z_2 \rangle$, then:

$\vec{v} \cdot \vec{w} = x_1x_2 + y_1y_2 + z_1z_2$

• For 2D vectors $\vec{v} = \langle x_1, y_1 \rangle$ and $\vec{w} = \langle x_2, y_2 \rangle$:

$\vec{v} \cdot \vec{w} = x_1x_2 + y_1y_2$

Properties of the Dot Product

• Commutativity: $\vec{v} \cdot \vec{w} = \vec{w} \cdot \vec{v}$

• Scalar Associativity: $c(\vec{v} \cdot \vec{w}) = (c\vec{v}) \cdot \vec{w} = \vec{v} \cdot (c\vec{w})$

• Result is a Scalar: The dot product always yields a scalar value.

• Dimension Requirement: Only vectors of the same dimension can be dotted.

Relation to Vector Length

• The dot product of a vector with itself gives the square of its magnitude:

$\vec{v} \cdot \vec{v} = x^2 + y^2 + z^2 = |\vec{v}|^2$

Geometric Interpretation

• The dot product can also be expressed in terms of the angle $\theta$ between the vectors:

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

• If $\theta = \frac{\pi}{2}$ (90 degrees), then $\cos(\theta) = 0$ and $\vec{v} \cdot \vec{w} = 0$; the vectors are orthogonal (perpendicular).

Examples

• Given $\vec{v} = \langle 2, 1 \rangle$, $\vec{w} = \langle 3, -1 \rangle$:

$\vec{v} \cdot \vec{w} = 2 \cdot 3 + 1 \cdot (-1) = 6 - 1 = 5$

• Given $\vec{v} = \langle 3, 1, 3 \rangle$, $\vec{w} = \langle 0, -4, -1 \rangle$:

$\vec{v} \cdot \vec{w} = 3 \cdot 0 + 1 \cdot (-4) + 3 \cdot (-1) = 0 - 4 - 3 = -7$

• Orthogonal vectors: $\vec{v} = 2\hat{i}$, $\vec{w} = 4\hat{j}$, $\vec{v} \cdot \vec{w} = 0$

Finding the Angle Between Vectors

• Given $\vec{v} = \langle 2, 1 \rangle$, $\vec{w} = \langle 3, -1 \rangle$:

$\vec{v} \cdot \vec{w} = 5$ $|\vec{v}| = \sqrt{5}$ $|\vec{w}| = \sqrt{10}$ $\cos(\theta) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}| |\vec{w}|} = \frac{5}{\sqrt{5} \sqrt{10}} = \frac{1}{\sqrt{2}}$ $\theta = \frac{\pi}{4}$

Vector Projections

Scalar Projection

The scalar projection of $\vec{w}$ onto $\vec{v}$ is the length of the shadow of $\vec{w}$ in the direction of $\vec{v}$.

$\text{Scal}_{\vec{v}} \vec{w} = |\vec{w}| \cos(\theta) = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}|}$

Vector Projection

The vector projection of $\vec{w}$ onto $\vec{v}$ gives both the magnitude and direction:

$\text{proj}_{\vec{v}} \vec{w} = \frac{\vec{v} \cdot \vec{w}}{|\vec{v}|^2} \vec{v}$

• Example: Project $\vec{w} = \langle 2, -3, 6 \rangle$ onto $\vec{v} = \langle -1, 4, -4 \rangle$:

$\text{proj}_{\vec{v}} \vec{w} = \frac{-14}{49} \langle -1, 4, -4 \rangle = \langle \frac{14}{49}, -\frac{56}{49}, \frac{56}{49} \rangle$ Additional info: The calculation uses the formula above, with $\vec{v} \cdot \vec{w} = -14$ and $|\vec{v}|^2 = 49$.

Projection onto Basis Vectors

Projecting a vector $\vec{v} = \langle a, b, c \rangle$ onto the standard basis vectors $\hat{i}, \hat{j}, \hat{k}$ gives its coordinates:

• $\text{proj}_{\hat{i}} \vec{v} = a\hat{i} = \langle a, 0, 0 \rangle$

• $\text{proj}_{\hat{j}} \vec{v} = b\hat{j} = \langle 0, b, 0 \rangle$

• $\text{proj}_{\hat{k}} \vec{v} = c\hat{k} = \langle 0, 0, c \rangle$

Remarks on Projections

• Projecting onto $\hat{i}, \hat{j}, \hat{k}$ gives the decomposition of the vector into its coordinate directions.

• Projections are useful for finding components and resolving vectors in physics and engineering.

Summary Table: Dot Product Properties

Property	Description
Commutativity	$\vec{v} \cdot \vec{w} = \vec{w} \cdot \vec{v}$
Scalar Associativity	$c(\vec{v} \cdot \vec{w}) = (c\vec{v}) \cdot \vec{w} = \vec{v} \cdot (c\vec{w})$
Orthogonality	$\vec{v} \cdot \vec{w} = 0$ if $\vec{v}$ and $\vec{w}$ are perpendicular
Relation to Magnitude	$\vec{v} \cdot \vec{v} = |\vec{v}|^2$
Angle Formula	$\vec{v} \cdot \vec{w} = |\vec{v}| |\vec{w}| \cos(\theta)$

Applications

• Dot product is used to determine orthogonality, calculate work in physics, and project vectors onto other vectors.

• Projections are essential in decomposing forces, velocities, and other vector quantities in science and engineering.