Problem 9 - Advanced Engineering Mathematics

Question

Find (a) \(4 \mathrm{a}-2 \mathrm{~b}\) and \((\mathrm{b})-3 \mathrm{a}-5 \mathrm{~b}\). \(\mathbf{a}=\langle 1,-3\rangle, \mathbf{b}=\langle-1,1\rangle\)

Step-by-Step Solution

Answer

\( 4 \mathbf{a} - 2 \mathbf{b} = \langle 6, -14 \rangle \) and \( -3 \mathbf{a} - 5 \mathbf{b} = \langle -8, 14 \rangle \).

Step 1: Understand the Vectors

We are given two vectors: \( \mathbf{a} = \langle 1, -3 \rangle \) and \( \mathbf{b} = \langle -1, 1 \rangle \). We need to find \( 4 \mathbf{a} - 2 \mathbf{b} \) and \( -3 \mathbf{a} - 5 \mathbf{b} \).

Step 2: Find 4a

First, we calculate \( 4 \mathbf{a} \). Using \( \mathbf{a} = \langle 1, -3 \rangle \), multiply each component by 4: \[ 4 \mathbf{a} = 4 \times \langle 1, -3 \rangle = \langle 4, -12 \rangle \].

Step 3: Find 2b

Next, calculate \( 2 \mathbf{b} \). With \( \mathbf{b} = \langle -1, 1 \rangle \), multiply each component by 2:\[ 2 \mathbf{b} = 2 \times \langle -1, 1 \rangle = \langle -2, 2 \rangle \].

Step 4: Calculate 4a - 2b

Subtract \( 2 \mathbf{b} \) from \( 4 \mathbf{a} \):\[ 4 \mathbf{a} - 2 \mathbf{b} = \langle 4, -12 \rangle - \langle -2, 2 \rangle = \langle 4 + 2, -12 - 2 \rangle = \langle 6, -14 \rangle \].

Step 5: Find -3a

Calculate \( -3 \mathbf{a} \). Multiply each component of \( \mathbf{a} = \langle 1, -3 \rangle \) by -3:\[-3 \mathbf{a} = -3 \times \langle 1, -3 \rangle = \langle -3, 9 \rangle \].

Step 6: Find -5b

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

Step 7: Calculate -3a - 5b

Subtract \( 5 \mathbf{b} \) from \( -3 \mathbf{a} \):\[-3 \mathbf{a} - 5 \mathbf{b} = \langle -3, 9 \rangle - \langle 5, -5 \rangle = \langle -3 - 5, 9 + 5 \rangle = \langle -8, 14 \rangle \].

Key Concepts

Vector Addition

When dealing with vectors, addition is a fundamental operation. It involves combining two or more vectors to create a new vector. Vector addition is performed by simply adding the corresponding components of the vectors together. For instance, if we have two vectors, \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), their addition is performed as follows:
\[ \mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle \]
This means the first components of both vectors are added together, and then the second components are summed. Vector addition is useful in many real-world applications, including physics and engineering. It allows for the combination of quantities like displacement, velocity, or force.

• Each vector must have the same number of components to be added correctly.
• Vector addition results in another vector.

In the original exercise, vector addition was part of calculating \(4 \mathbf{a} - 2\mathbf{b}\), where terms were subtracted after first performing multiplication.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector by altering its magnitude, but not its direction unless the scalar is negative. Consider a vector \( \mathbf{v} = \langle x, y \rangle \) and a scalar \( k \). The scalar multiplication is performed by multiplying each component of \( \mathbf{v} \) by \( k \):
\[ k \mathbf{v} = k \times \langle x, y \rangle = \langle kx, ky \rangle \]
This operation outputs a new vector that points in the same direction as the original (if \( k > 0 \)), but is \( k \) times as long. If \( k < 0 \), the direction is reversed. An important point to remember is that scalar multiplication impacts magnitude but not component-wise relationships within a vector.

• Does not change direction unless the scalar is negative.
• Scaling affects the length or magnitude of the vector.

In the task at hand, scalar multiplication was crucial to scale \( \mathbf{a} \) and \( \mathbf{b} \) before performing subtraction.

Vector Subtraction

Vector subtraction is similar to vector addition, except it involves deducting the components of one vector from another. This operation can be thought of as adding a negative vector. For vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), vector subtraction is given by:
\[ \mathbf{u} - \mathbf{v} = \langle u_1 - v_1, u_2 - v_2 \rangle \]
Here, each component of the second vector is subtracted from the corresponding component of the first vector. Vector subtraction is useful in determining differences in quantities like displacement or velocity between two points.

• First negate the vector you want to subtract and then perform an addition.
• Resulting vector indicates how to go from the tip of one vector to another.

In the original solution, calculating expressions like \(4 \mathbf{a} - 2 \mathbf{b}\) and \(-3 \mathbf{a} - 5 \mathbf{b} \), involved calculation of scaled vectors followed by vector subtraction.