Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 23-28, find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{v}} \mathbf{u} $$

Step-by-Step Solution

Answer

The projection of \( \mathbf{u} \) onto \( \mathbf{v} \) is the zero vector \( \mathbf{0} \).

Step 1: Understand the Projection Formula

To find the projection of vector \( \mathbf{u} \) onto vector \( \mathbf{v} \), we use the formula: \[ \operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \] where \( \mathbf{u} \cdot \mathbf{v} \) is the dot product of \( \mathbf{u} \) and \( \mathbf{v} \), and \( \mathbf{v} \cdot \mathbf{v} \) is the dot product of \( \mathbf{v} \) with itself.

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

Compute the dot product of \( \mathbf{u} \) and \( \mathbf{v} \): \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} - \mathbf{j} \). The dot product \( \mathbf{u} \cdot \mathbf{v} \) is calculated as: \[ (1)(2) + (2)(-1) = 2 - 2 = 0 \]

Step 3: Calculate the Dot Product \( \mathbf{v} \cdot \mathbf{v} \)

Compute the dot product of \( \mathbf{v} \) with itself: \( \mathbf{v} = 2\mathbf{i} - \mathbf{j} \). The dot product \( \mathbf{v} \cdot \mathbf{v} \) is given by: \[ (2)(2) + (-1)(-1) = 4 + 1 = 5 \]

Step 4: Calculate the Projection

Substitute the dot products into the projection formula: \[ \operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{0}{5} \mathbf{v} = 0 \cdot \mathbf{v} = \mathbf{0} \] Hence, the projection of \( \mathbf{u} \) onto \( \mathbf{v} \) is the zero vector.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It combines pairs of vectors to produce a single scalar quantity. This scalar represents the product of the magnitudes of the two vectors and the cosine of the angle between them. Mathematically, the dot product of two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) is defined as: \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \] In simple terms, each corresponding component of the vectors is multiplied and then summed up:

• \( a_1 \) is multiplied by \( b_1 \)
• \( a_2 \) is multiplied by \( b_2 \)

The dot product is particularly important because it determines, among other things, the angle relationship between two vectors. If the dot product is zero, the vectors are perpendicular. In our exercise with vectors \( \mathbf{u} = \mathbf{i} + 2 \mathbf{j} \) and \( \mathbf{v} = 2 \mathbf{i} - \mathbf{j} \), we found that their dot product is zero, indicating they are perpendicular.

Projection Formula

The projection formula helps us find the shadow of one vector onto another. It is used to determine the component of one vector in the direction of another. Using the projection formula for vector \( \mathbf{u} \) onto vector \( \mathbf{v} \), we have: \[ \operatorname{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \] Each term plays a crucial role in calculating the projection:

• \( \mathbf{u} \cdot \mathbf{v} \): Calculates the extent to which \( \mathbf{u} \) is in the direction of \( \mathbf{v} \)
• \( \mathbf{v} \cdot \mathbf{v} \): Scales the denominator to the length of \( \mathbf{v} \)
• \( \mathbf{v} \): Determines the direction

In the exercise, when substituting the values into this formula, it was found that the projection is the zero vector \( \mathbf{0} \). This resulted because the dot product \( \mathbf{u} \cdot \mathbf{v} = 0 \), leading to a zero projection.

Vectors in Mathematics

Vectors are a fundamental concept in mathematics, especially in geometry and physics. A vector is defined by its magnitude and direction and is often represented with an arrow, where the direction and length visually represent these properties. In common mathematical notation, vectors are expressed as combinations of unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \): \( \mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k} \). In two dimensions, vectors simplify to linear combinations of \( \mathbf{i} \) and \( \mathbf{j} \) as seen in our example vectors \( \mathbf{u} \) and \( \mathbf{v} \):

• \( \mathbf{u} = \mathbf{i} + 2 \mathbf{j} \)
• \( \mathbf{v} = 2 \mathbf{i} - \mathbf{j} \)

Understanding vectors' behavior helps to solve problems involving direction and magnitude. Well-known operations include the dot product, used in projections, and the cross product which applies in three dimensions. Vectors not only represent physical quantities like force or velocity but also find utility in more abstract fields such as computer graphics and systems of equations.