Problem 24 - Algebra and Trigonometry

Question

Two vectors \(\mathbf{u}\) and \(\mathbf{v}\) are given. Express the vector \(-2 \mathbf{u}+3 \mathbf{v}(\mathbf{a})\) in component form \(\left\langle a_{1}, a_{2}, a_{3}\right\rangle\) and \((\mathbf{b})\) in terms of $$ \mathbf{u}=\langle 3,1,0\rangle, \quad \mathbf{v}=\langle 3,0,-5\rangle $$

Step-by-Step Solution

Answer

The vector is \(\langle 3, -2, -15 \rangle\).

Step 1: Understand the Problem

We need to express the vector \(-2\mathbf{u} + 3\mathbf{v}\) in component form using the given vectors \(\mathbf{u}\) and \(\mathbf{v}\).

Step 2: Multiply Vector \(\mathbf{u}\) by Scalar

Given \(\mathbf{u} = \langle 3, 1, 0 \rangle\), multiply each component by \(-2\):\[-2\mathbf{u} = -2 \times \langle 3, 1, 0 \rangle = \langle -6, -2, 0 \rangle.\]

Step 3: Multiply Vector \(\mathbf{v}\) by Scalar

Given \(\mathbf{v} = \langle 3, 0, -5 \rangle\), multiply each component by \(3\):\[3\mathbf{v} = 3 \times \langle 3, 0, -5 \rangle = \langle 9, 0, -15 \rangle.\]

Step 4: Add Resulting Vectors

Add the resulting vectors \(-2\mathbf{u}\) and \(3\mathbf{v}\) component-wise:\[-2\mathbf{u} + 3\mathbf{v} = \langle -6, -2, 0 \rangle + \langle 9, 0, -15 \rangle = \langle -6 + 9, -2 + 0, 0 + (-15) \rangle.\]

Step 5: Calculate the Components

Complete the addition for each component:\[-6 + 9 = 3,\]\[-2 + 0 = -2,\]\[0 + (-15) = -15.\]So, the vector is \(\langle 3, -2, -15 \rangle\).

Key Concepts

Vector Addition

Vector addition is an essential operation used to combine two or more vectors to produce a new vector. Understanding how to correctly add vectors is fundamental in physics and engineering. It enables us to determine resultant vectors, which explain the cumulative effect of two or more vector quantities acting together.

When adding vectors, we do this component-wise. This means adding the corresponding components of each vector together. For example, if you have two vectors, \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), vector addition would result in a new vector:
\[ \mathbf{a} + \mathbf{b} = \langle a_1+b_1, a_2+b_2, a_3+b_3 \rangle. \]

• The first components are added together: \( a_1 + b_1 \).
• The second components are added together: \( a_2 + b_2 \).
• The third components are added together: \( a_3 + b_3 \).

Remember, the vectors being added must have the same dimensions, as shown in our problem with the vectors \(-2\mathbf{u}\) and \(3\mathbf{v}\). By adding them together, we obtained a new vector \(\langle 3, -2, -15 \rangle\).

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a single number, called a 'scalar'. This operation is vital because it allows you to adjust the magnitude of a vector without altering its direction, except when the scalar is negative, which flips its direction.

For the vectors in our exercise, the operation is straightforward. Consider vector \( \mathbf{u} = \langle 3, 1, 0 \rangle \) and the scalar \(-2\). To multiply \( \mathbf{u} \) by the scalar \(-2\), each component of \( \mathbf{u} \) is individually multiplied by \(-2\):
\[ -2\mathbf{u} = -2 \cdot \langle 3, 1, 0 \rangle = \langle -6, -2, 0 \rangle. \]

Likewise, for \( \mathbf{v} = \langle 3, 0, -5 \rangle \) and the scalar \(3\):
\[ 3\mathbf{v} = 3 \cdot \langle 3, 0, -5 \rangle = \langle 9, 0, -15 \rangle. \]

• Each component is scaled by multiplying with the scalar.
• A negative scalar inverses the direction of the vector.
• A larger scalar increases the vector's magnitude.

This operation scales the vector while maintaining its original directionality unless the scalar is negative.

Component Form

The component form of a vector is a way of expressing a vector using its individual parts or components. This makes working with vectors, especially in more complex problems, much more manageable.

A vector in component form is typically represented as \( \langle a_1, a_2, a_3, ..., a_n \rangle \), where each \( a_i \) is a component of the vector. By breaking vectors into components, you can easily perform operations like addition or scalar multiplication using simple arithmetic.

In our exercise, vectors \( \mathbf{u} \) and \( \mathbf{v} \) are already given in component form:

• \( \mathbf{u} = \langle 3, 1, 0 \rangle \)
• \( \mathbf{v} = \langle 3, 0, -5 \rangle \)

Knowing the component form allows us to easily multiply each part by a scalar and add corresponding components from two vectors systematically.

Thus, the result of \(-2\mathbf{u} + 3\mathbf{v}\) is also expressed in component form as \( \langle 3, -2, -15 \rangle \). The neat organization of vector components ensures clarity and facilitates vector operations across various applications.