Problem 6 - Precalculus: Functions and Graphs

Question

Find \(a+b, a-b, 4 a+5 b, 4 a-5 b,\) and \(\|a\|\) $$\mathbf{a}=-3 \mathbf{i}+\mathbf{j}, \quad \mathbf{b}=-3 \mathbf{i}+\mathbf{j}$$

Step-by-Step Solution

Answer

\( \mathbf{a} + \mathbf{b} = -6\mathbf{i} + 2\mathbf{j} \), \( \mathbf{a} - \mathbf{b} = 0 \), \( 4\mathbf{a} + 5\mathbf{b} = -27\mathbf{i} + 9\mathbf{j} \), \( 4\mathbf{a} - 5\mathbf{b} = 3\mathbf{i} - \mathbf{j} \), \( \|\mathbf{a}\| = \sqrt{10} \).

Step 1: Understand the Vectors

The vectors given are \( \mathbf{a} = -3 \mathbf{i} + \mathbf{j} \) and \( \mathbf{b} = -3 \mathbf{i} + \mathbf{j} \). These vectors are expressed in terms of their components along the \( \mathbf{i} \) (horizontal) and \( \mathbf{j} \) (vertical) axes.

Step 2: Calculate \( \mathbf{a} + \mathbf{b} \)

For \( \mathbf{a} + \mathbf{b} \), add corresponding components of \( \mathbf{a} \) and \( \mathbf{b} \):\[ (-3 + -3)\mathbf{i} + (1 + 1)\mathbf{j} = -6\mathbf{i} + 2\mathbf{j} \].

Step 3: Calculate \( \mathbf{a} - \mathbf{b} \)

For \( \mathbf{a} - \mathbf{b} \), subtract components of \( \mathbf{b} \) from components of \( \mathbf{a} \):\[ (-3 - (-3))\mathbf{i} + (1 - 1)\mathbf{j} = 0\mathbf{i} + 0\mathbf{j} \].

Step 4: Calculate \( 4\mathbf{a} + 5\mathbf{b} \)

Multiply each component of \( \mathbf{a} \) and \( \mathbf{b} \) by their respective scalars, then add:- For \( 4\mathbf{a} \): \( 4(-3)\mathbf{i} + 4(1)\mathbf{j} = -12\mathbf{i} + 4\mathbf{j} \).- For \( 5\mathbf{b} \): \( 5(-3)\mathbf{i} + 5(1)\mathbf{j} = -15\mathbf{i} + 5\mathbf{j} \).- Add these results: \( (-12 - 15)\mathbf{i} + (4 + 5)\mathbf{j} = -27\mathbf{i} + 9\mathbf{j} \).

Step 5: Calculate \( 4\mathbf{a} - 5\mathbf{b} \)

Multiply each component of \( \mathbf{a} \) and \( \mathbf{b} \) by their respective scalars, then subtract component-wise:- For \( 4\mathbf{a} \): \( -12\mathbf{i} + 4\mathbf{j} \) (as calculated previously).- For \( 5\mathbf{b} \): \( -15\mathbf{i} + 5\mathbf{j} \) (as calculated previously).- Subtract these results: \( (-12 + 15)\mathbf{i} + (4 - 5)\mathbf{j} = 3\mathbf{i} - \mathbf{j} \).

Step 6: Calculate the Magnitude of \( \|\mathbf{a}\| \)

The magnitude \( \|\mathbf{a}\| \) is calculated using the formula:\[ \| \mathbf{a} \| = \sqrt{(-3)^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \].

Key Concepts

Vector Addition

Adding vectors is all about combining them to find a resultant vector. In the case of vectors \( \mathbf{a} = -3\mathbf{i} + \mathbf{j} \) and \( \mathbf{b} = -3\mathbf{i} + \mathbf{j} \), vector addition involves summing their corresponding components. For these specific vectors, we add the \( \mathbf{i} \) components separately from the \( \mathbf{j} \) components. Here’s how it works:

• The \( \mathbf{i} \) part: \((-3) + (-3) = -6\)
• The \( \mathbf{j} \) part: \(1 + 1 = 2\)

So, \( \mathbf{a} + \mathbf{b} = -6\mathbf{i} + 2\mathbf{j} \). Add vector components to get a new vector, representing both forces or directions combined. This method applies to vectors in all dimensions.

Vector Subtraction

Subtracting vectors is the process of finding the difference between two vectors, often visualized as the vector pointing from the tip of one vector to the tip of the other. For vectors \( \mathbf{a} = -3\mathbf{i} + \mathbf{j} \) and \( \mathbf{b} = -3\mathbf{i} + \mathbf{j} \), vector subtraction takes place component-wise as follows:

• Subtract the \( \mathbf{i} \) components: \((-3) - (-3) = 0\)
• Subtract the \( \mathbf{j} \) components: \(1 - 1 = 0\)

Therefore, \( \mathbf{a} - \mathbf{b} = 0\mathbf{i} + 0\mathbf{j} \), which is simply the zero vector. This shows that both vectors point in the same direction and are of equal length.

Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a constant, known as the scalar. This operation changes the magnitude of the vector without altering its direction, unless the scalar is negative, which reverses the direction. For \( 4\mathbf{a} + 5\mathbf{b} \) and \( 4\mathbf{a} - 5\mathbf{b} \), you perform these calculations:

1. Multiply the vector \( \mathbf{a} = -3\mathbf{i} + \mathbf{j} \) by 4:

• \( 4(-3\mathbf{i}) = -12\mathbf{i} \)
• \( 4(\mathbf{j}) = 4\mathbf{j} \)

2. Multiply the vector \( \mathbf{b} = -3\mathbf{i} + \mathbf{j} \) by 5:

• \( 5(-3\mathbf{i}) = -15\mathbf{i} \)
• \( 5(\mathbf{j}) = 5\mathbf{j} \)

Then, either add or subtract these results to complete combining vectors under multiplication to effectively change their size before combining.

Magnitude of a Vector

The magnitude of a vector is a measure of its length or size. For a vector expressed as \( \mathbf{a} = -3\mathbf{i} + \mathbf{j} \), calculate the magnitude using the Pythagorean theorem, which applies by treating the vector components as sides of a right triangle. The magnitude \( \|\mathbf{a}\| \) is found by:

• Squaring each component: \((-3)^2 = 9\) and \(1^2 = 1\)
• Adding the squares: \(9 + 1 = 10\)
• Taking the square root: \( \sqrt{10} \)

Hence, \( \|\mathbf{a}\| = \sqrt{10} \), revealing how far the vector stretches from its starting point, which is helpful in physics when considering forces or velocities.