Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the scalar projection of \(\mathbf{u}=-\mathbf{i}+5 \mathbf{j}+3 \mathbf{k}\) on \(\mathbf{v}=-\mathbf{i}+\mathbf{j}-\mathbf{k}\).

Step-by-Step Solution

Answer

The scalar projection of \( \mathbf{u} \) on \( \mathbf{v} \) is \( \sqrt{3} \).

Step 1: Understand Scalar Projection Concept

The scalar projection of a vector \( \mathbf{u} \) onto another vector \( \mathbf{v} \) is given by the formula: \( \text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{\mathbf{u} \cdot \mathbf{v}}{\| \mathbf{v} \|} \). This means the scalar projection is the length of the shadow of \( \mathbf{u} \) when it's cast onto \( \mathbf{v} \).

Step 2: Calculate the Dot Product \(\mathbf{u} \cdot \mathbf{v}\)

First, calculate the dot product of \( \mathbf{u} = -\mathbf{i} + 5\mathbf{j} + 3\mathbf{k} \) and \( \mathbf{v} = -\mathbf{i} + \mathbf{j} - \mathbf{k} \). The dot product is given by: \( \mathbf{u} \cdot \mathbf{v} = (-1)(-1) + (5)(1) + (3)(-1) = 1 + 5 - 3 = 3 \).

Step 3: Calculate the Magnitude \(\| \mathbf{v} \|\)

Calculate the magnitude of \( \mathbf{v} = -\mathbf{i} + \mathbf{j} - \mathbf{k} \). Use the formula: \( \| \mathbf{v} \| = \sqrt{(-1)^2 + (1)^2 + (-1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3} \).

Step 4: Compute Scalar Projection

Substitute the dot product and magnitude into the scalar projection formula: \( \text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{3}{\sqrt{3}} = \sqrt{3} \).

Key Concepts

Scalar Projection

The concept of a scalar projection is somewhat like measuring how much one vector extends in the direction of another vector. Think of shining a light directly above vector \( \mathbf{u} \) onto vector \( \mathbf{v} \). The shadow that appears on \( \mathbf{v} \) is what the scalar projection represents. It tells us how one vector 'projects' onto the other in terms of length.
To compute it, we use the formula \( \text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{\mathbf{u} \cdot \mathbf{v}}{\| \mathbf{v} \|} \). This formula requires two components: the dot product of the vectors and the magnitude of the vector you are projecting onto.
The result of this calculation is a single number (or scalar), giving the length of the projection. It is essential in physics and engineering where direction matters as much as the magnitude.

Dot Product

The dot product, also known as the scalar product, is one of the two types of multiplications that can be performed on vectors. It is a way of multiplying two vectors to obtain a scalar. The dot product quantifies how much one vector extends in the direction of another.
To find the dot product \( \mathbf{u} \cdot \mathbf{v} \) of two vectors \( \mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j} + u_3\mathbf{k} \) and \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k} \), calculate this:

• Multiply the corresponding components: \( u_1v_1, u_2v_2, u_3v_3 \).
• Add the products together: \( u_1v_1 + u_2v_2 + u_3v_3 \).

For example, if \( \mathbf{u} = -\mathbf{i} + 5\mathbf{j} + 3\mathbf{k} \) and \( \mathbf{v} = -\mathbf{i} + \mathbf{j} - \mathbf{k} \), the dot product will be calculated as \( 1 + 5 - 3 = 3 \).
This results in a scalar value that serves as the numerator in the scalar projection formula.

Vector Magnitude

Vector magnitude is an indication of the length or size of a vector. Understanding magnitude is crucial when working with projections, as it represents how long the vector extends in space.
To calculate the magnitude of a vector \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k} \), we use the formula \( \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2 + v_3^2} \). This is similar to finding the length of the hypotenuse in a right triangle.
In our example with \( \mathbf{v} = -\mathbf{i} + \mathbf{j} - \mathbf{k} \), the components are \( -1, 1, \) and \( -1 \). Thus,

• Square each component: \((-1)^2 = 1\), \((1)^2 = 1\), \((-1)^2 = 1\).
• Add them up: \(1 + 1 + 1 = 3\).
• Take the square root of the sum: \(\sqrt{3}\).

The magnitude, a scalar itself, plays a key role in normalizing direction in vector calculations.