Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each of the given projections if \(\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{v}=2 \mathbf{i}-\mathbf{k}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{i}} \mathbf{u}\)

Step-by-Step Solution

Answer

\(\operatorname{proj}_{\mathbf{i}} \mathbf{u} = 3 \mathbf{i}\)

Step 1: Find Dot Product of u and i

First, identify the unit vector \(\mathbf{i}\), which is \((1, 0, 0)\). Calculate the dot product of \(\mathbf{u}\) and \(\mathbf{i}\) using the formula \(\mathbf{u} \cdot \mathbf{i} = 3 \times 1 + 2 \times 0 + 1 \times 0 = 3\).

Step 2: Calculate Magnitude of i

The magnitude of the unit vector \(\mathbf{i}\) is 1, as it is a unit vector. Thus, \(\|\mathbf{i}\| = 1\).

Step 3: Compute Projection Formula

The formula for the projection of \(\mathbf{u}\) onto \(\mathbf{i}\) is given by \(\operatorname{proj}_{\mathbf{i}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{i}}{\|\mathbf{i}\|^2} \mathbf{i}\). Since the magnitude of \(\mathbf{i}\) is 1, \(\|\mathbf{i}\|^2 = 1\).

Step 4: Calculate the Projection

Substitute the values into the projection formula: \(\operatorname{proj}_{\mathbf{i}} \mathbf{u} = \frac{3}{1} \cdot \mathbf{i} = 3 \mathbf{i}\).

Key Concepts

Dot Product

The dot product is a fundamental concept in vector mathematics. It is essentially a way to multiply two vectors to get a scalar (a single number) result. To calculate the dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\), you follow this formula:

• Multiply each corresponding component of the vectors.
• Add those products together.

For example, if you have vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\), the dot product is expressed as: \(\mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3\).This concept is pivotal when finding projections, as the dot product helps determine how much of one vector goes in the direction of another.

Unit Vector

A unit vector is a vector that has a magnitude of one. It is used to specify directions without concerning the scale.

• Mathematically, a unit vector in a given direction can be computed by dividing any non-zero vector by its magnitude.
• Common unit vectors in the Cartesian coordinate system include \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), representing directions along the x, y, and z axes, respectively.

For instance, \(\mathbf{i} = (1, 0, 0)\) is a unit vector that points purely in the direction of the x-axis. The role of unit vectors becomes clear when they help in scaling projections and aligning vectors in specific directions with ease.

Magnitude

The magnitude of a vector quantifies its length or size. Think of it like the vector's total displacement from the origin when plotted in space.To find the magnitude of a vector \(\mathbf{a} = (a_1, a_2, a_3)\), use the formula:\[ \|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]

• This formula is derived from the Pythagorean theorem, extended into three dimensions for a 3D space.
• With unit vectors like \(\mathbf{i}, \mathbf{j}, \mathbf{k}\), their magnitude is always 1, simplifying many calculations.

Understanding magnitude is key to knowing how vectors compare in size and assessing their influence in different directions.

Projection Formula

The projection formula allows us to "project" one vector onto another, essentially finding a component of the vector in the direction of the other.The formula for projecting \(\mathbf{u}\) onto \(\mathbf{v}\) (where \(\mathbf{v}\) is the vector we project onto) is:\[\operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v}\]Here's a breakdown:

• The dot product \(\mathbf{u} \cdot \mathbf{v}\) measures the extent to which \(\mathbf{u}\) is in the direction of \(\mathbf{v}\).
• \(\|\mathbf{v}\|^2\) in the denominator scales the projection appropriately, involving the length of \(\mathbf{v}\).
• Multiplying by \(\mathbf{v}\) ensures the resulting vector is in the direction of \(\mathbf{v}\).

This formula enables us to extract and utilize information on directional influence between vectors.