Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The projection of \( \mathbf{u} \) onto \( \mathbf{i} \) is \( \mathbf{i} \).

Step 1: Identify the Formula

The projection of vector \( \mathbf{u} \) onto a direction vector \( \mathbf{a} \) is given by \( \operatorname{proj}_{\mathbf{a}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{a}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a} \). Here, we want the projection onto \( \mathbf{i} \), which becomes \( \operatorname{proj}_{\mathbf{i}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{i}}{\mathbf{i} \cdot \mathbf{i}} \mathbf{i} \).

Step 2: Calculate Dot Products

The dot product \( \mathbf{u} \cdot \mathbf{i} \) is \( (1)\cdot(1) + (2)\cdot(0) = 1 \). The dot product \( \mathbf{i} \cdot \mathbf{i} \) is \( (1)\cdot(1) + (0)\cdot(0) = 1 \).

Step 3: Compute Projection

Using the formula \( \operatorname{proj}_{\mathbf{i}} \mathbf{u} = \frac{1}{1} \mathbf{i} \), simplify to find the projection to be \( \operatorname{proj}_{\mathbf{i}} \mathbf{u} = \mathbf{i} \).

Key Concepts

Dot Product

The dot product is an essential operation in vector calculus and linear algebra, and it's crucial for finding vector projections. It's a way to multiply two vectors that results in a scalar, rather than another vector.

When you have two vectors, say \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), the dot product is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \).

• This operation is commutative, meaning \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).
• The result of a dot product can tell us about the relationship between two vectors, such as their angle and whether they are perpendicular (dot product equals zero).

Understanding the dot product is key to grasping vector projections, as it helps project one vector onto another.

Basis Vectors

Basis vectors form the fundamental building blocks of vector spaces in linear algebra. They are usually denoted as \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \) in three-dimensional space.

These vectors are crucial because they provide a way to describe any vector in the space through linear combinations:

• For example, in two dimensions, any vector \( \mathbf{u} = x \mathbf{i} + y \mathbf{j} \).
• The vectors \( \mathbf{i} = (1,0) \) and \( \mathbf{j} = (0,1) \) create a standard basis for 2D space.

Each vector's components represent how much \( \mathbf{i} \) and \( \mathbf{j} \) are needed to reach it from the origin. Mastering basis vectors helps in understanding concepts like vector projections, since projections often occur onto basis vectors themselves.

Vector Calculus

Vector calculus extends the principles of calculus to vector functions. It incorporates operations like differentiation and integration but applied to vector quantities.

Here's why it's useful in the study of vector projections:

• Vector-valued functions can represent paths or trajectories.
• The derivatives can represent velocity and acceleration vectors along those paths.
• When projecting one vector onto another, you're inherently involving calculus-type thinking, as you study vector components and their directions.

Vector calculus allows for deeper understanding in fields that rely heavily on vectors, such as physics and engineering, providing a framework to solve real-world problems more effectively.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors and linear transformations. It's the mathematical toolkit behind vector calculus.

Linear algebra covers topics such as:

• Vectors: Both row and column vectors, and their operations.
• Matrices: Arrays of numbers that can represent linear transformations.
• Systems of linear equations: Solving these systems using vectors and matrices.

In this context, vector projections are understood as one vector being expressed in terms of another, often using a basis. This is crucial for calculations involving coordinates, where expressing a vector in terms of orthogonal components can simplify complex problems. The interplay of these concepts, grounded in linear algebra, empowers solving many practical and theoretical problems.