Problem 21 - Thomas Calculus

Question

Express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k}.\) \(5 \mathbf{u}-\mathbf{v}\) if \(\mathbf{u}=\langle 1,1,-1\rangle\) and \(\mathbf{v}=\langle 2,0,3\rangle\)

Step-by-Step Solution

Answer

\(5 \mathbf{u} - \mathbf{v} = 3 \mathbf{i} + 5 \mathbf{j} - 8 \mathbf{k}\).

Step 1: Understand the Problem

We are asked to express a new vector, \(5 \mathbf{u} - \mathbf{v}\), using the components \((v_1, v_2, v_3)\) in the standard unit vector form \(\mathbf{v}=v_{1} \mathbf{i} + v_{2} \mathbf{j} + v_{3} \mathbf{k}\). Given vectors \(\mathbf{u} = \langle 1, 1, -1 \rangle\) and \(\mathbf{v} = \langle 2, 0, 3 \rangle\), we will perform scalar multiplication and vector subtraction.

Step 2: Scalar Multiplication

Calculate \(5 \mathbf{u}\). Multiply each component of \(\mathbf{u}\) by 5: \[ 5 \mathbf{u} = 5 \langle 1, 1, -1 \rangle = \langle 5(1), 5(1), 5(-1) \rangle = \langle 5, 5, -5 \rangle. \]

Step 3: Vector Subtraction

Subtract \(\mathbf{v}\) from \(5 \mathbf{u}\). Subtract each corresponding component of \(\mathbf{v}\) from \(5 \mathbf{u}\): \[ 5 \mathbf{u} - \mathbf{v} = \langle 5, 5, -5 \rangle - \langle 2, 0, 3 \rangle = \langle (5-2), (5-0), (-5-3) \rangle = \langle 3, 5, -8 \rangle. \]

Step 4: Express in Unit Vector Form

Convert the resulting vector \(\langle 3, 5, -8 \rangle\) into the unit vector form: \[ 5 \mathbf{u} - \mathbf{v} = 3 \mathbf{i} + 5 \mathbf{j} - 8 \mathbf{k}. \]

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental concept in vector algebra where each component of a vector is multiplied by a single number called a scalar. In this concept, the magnitude of the vector changes, but its direction remains the same if the scalar is positive. If the scalar is negative, the vector's direction is reversed. This operation allows us to scale a vector by making it longer, shorter, or reversing its direction.
To grasp this, consider the vector \( \mathbf{u} = \langle 1, 1, -1 \rangle \). To perform scalar multiplication by 5, you multiply each of the vector’s components by 5:

\[ 5 \mathbf{u} = 5 \langle 1, 1, -1 \rangle = \langle 5 \times 1, 5 \times 1, 5 \times (-1) \rangle = \langle 5, 5, -5 \rangle \]
This operation is straightforward, yet crucial for manipulating vectors in various applications, from geometry to physics.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors by subtracting corresponding components. It is akin to vector addition, but instead, we add the opposite of the second vector.
For example, consider vectors \( 5 \mathbf{u} = \langle 5, 5, -5 \rangle \) and \( \mathbf{v} = \langle 2, 0, 3 \rangle \). To subtract \( \mathbf{v} \) from \( 5 \mathbf{u} \), compute the difference for each component:

• Subtract the first components: \( 5 - 2 = 3 \)
• Subtract the second components: \( 5 - 0 = 5 \)
• Subtract the third components: \( -5 - 3 = -8 \)

Thus, the resulting vector is \( \langle 3, 5, -8 \rangle \).

This operation allows you to determine the vector that represents moving from the endpoint of the second vector to the endpoint of the first vector. It's a vital operation for calculating relative positions and directions.

Unit Vector Form

In vector notation, expressing a vector in unit vector form means presenting it as a combination of the standard basis vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), which align with the x, y, and z axes, respectively. These basis vectors have a magnitude of one and are used to simplify the representation of vectors in three-dimensional space.
To convert a vector \( \langle 3, 5, -8 \rangle \) into unit vector form, each component is multiplied by its respective unit vector:

\[ 5 \mathbf{u} - \mathbf{v} = 3 \mathbf{i} + 5 \mathbf{j} - 8 \mathbf{k} \]
This form is beneficial because it provides a clear and consistent method of expressing vectors.

• \( \mathbf{i} \) is the unit vector parallel to the x-axis.
• \( \mathbf{j} \) is the unit vector parallel to the y-axis.
• \( \mathbf{k} \) is the unit vector parallel to the z-axis.

Unit vector form is widely used in physics and engineering, providing a convenient way to handle calculations involving direction and magnitude.