7.2 Q-6

Question Statement

Given the vectors:

• $\underline{\mathbf{a}} = 3 \underline{\mathbf{i}} - \underline{\mathbf{j}} - 4 \underline{\mathbf{k}}$
• $\underline{\mathbf{b}} = -2 \underline{\mathbf{i}} - 4 \underline{\mathbf{j}} - 3 \underline{\mathbf{k}}$
• $\underline{\mathbf{c}} = \underline{\mathbf{i}} + 2 \underline{\mathbf{j}} - \underline{\mathbf{k}}$

Find a unit vector parallel to the vector: $\mathbf{3} \underline{a} - 2 \underline{b} + 4 \underline{c}$

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Background and Explanation

To solve this problem, we need to:

1. Calculate the resultant vector by applying the operations on the given vectors.
2. Find the magnitude of the resultant vector.
3. Convert the resultant vector into a unit vector by dividing each component of the vector by its magnitude.

A unit vector is a vector with a magnitude of 1, and it points in the same direction as the original vector. To find a unit vector, we divide the vector by its magnitude: $\hat{d} = \frac{\mathbf{d}}{|\mathbf{d}|}$

Where:

• $\mathbf{d}$ is the resultant vector.
• $|\mathbf{d}|$ is the magnitude of the vector $\mathbf{d}$.

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Solution

Step 1: Find the resultant vector $\mathbf{d} = 3 \underline{a} - 2 \underline{b} + 4 \underline{c}$

Start by calculating $3 \underline{a}$, $-2 \underline{b}$, and $4 \underline{c}$.

1. Calculate $3 \underline{a}$: $3 \underline{a} = 3(3 \underline{i} - \underline{j} - 4 \underline{k}) = 9 \underline{i} - 3 \underline{j} - 12 \underline{k}$

2. Calculate $-2 \underline{b}$: $-2 \underline{b} = -2(-2 \underline{i} - 4 \underline{j} - 3 \underline{k}) = 4 \underline{i} + 8 \underline{j} + 6 \underline{k}$

3. Calculate $4 \underline{c}$: $4 \underline{c} = 4(\underline{i} + 2 \underline{j} - \underline{k}) = 4 \underline{i} + 8 \underline{j} - 4 \underline{k}$

Now, combine all the terms to find the resultant vector $\mathbf{d}$: $\mathbf{d} = (9 \underline{i} - 3 \underline{j} - 12 \underline{k}) + (4 \underline{i} + 8 \underline{j} + 6 \underline{k}) + (4 \underline{i} + 8 \underline{j} - 4 \underline{k})$

Combine the like components:

• For $\mathbf{i}$: $9 + 4 + 4 = 17 \underline{i}$
• For $\mathbf{j}$: $-3 + 8 + 8 = 13 \underline{j}$
• For $\mathbf{k}$: $-12 + 6 - 4 = -10 \underline{k}$

Thus, the resultant vector is: $\mathbf{d} = 17 \underline{i} + 13 \underline{j} - 10 \underline{k}$

Step 2: Find the magnitude of $\mathbf{d}$

The magnitude of a vector $\mathbf{d} = d_x \underline{i} + d_y \underline{j} + d_z \underline{k}$ is given by: $|\mathbf{d}| = \sqrt{d_x^2 + d_y^2 + d_z^2}$

Substitute the components of $\mathbf{d}$: $|\mathbf{d}| = \sqrt{(17)^2 + (13)^2 + (-10)^2}$

Calculate each term: $|\mathbf{d}| = \sqrt{289 + 169 + 100} = \sqrt{558}$

So, the magnitude of $\mathbf{d}$ is: $|\mathbf{d}| = \sqrt{558}$

Step 3: Find the unit vector in the direction of $\mathbf{d}$

Now, we find the unit vector by dividing each component of $\mathbf{d}$ by its magnitude $|\mathbf{d}|$: $\hat{\mathbf{d}} = \frac{\mathbf{d}}{|\mathbf{d}|} = \frac{17 \underline{i} + 13 \underline{j} - 10 \underline{k}}{\sqrt{558}}$

Thus, the unit vector is: $\hat{\mathbf{d}} = \frac{17}{\sqrt{558}} \underline{i} + \frac{13}{\sqrt{558}} \underline{j} - \frac{10}{\sqrt{558}} \underline{k}$

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Key Formulas or Methods Used

• Magnitude of a vector: $|\mathbf{d}| = \sqrt{d_x^2 + d_y^2 + d_z^2}$

• Unit vector: $\hat{\mathbf{d}} = \frac{\mathbf{d}}{|\mathbf{d}|}$

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Summary of Steps

1. Find the components of the vector $\mathbf{d} = 3 \underline{a} - 2 \underline{b} + 4 \underline{c}$ by performing the scalar multiplication and vector addition.
2. Calculate the magnitude of the resultant vector $\mathbf{d}$ using the formula for the magnitude of a vector.
3. Find the unit vector by dividing each component of $\mathbf{d}$ by its magnitude.
4. The unit vector parallel to $\mathbf{3} \underline{a} - 2 \underline{b} + 4 \underline{c}$ is: $\hat{\mathbf{d}} = \frac{17}{\sqrt{558}} \underline{i} + \frac{13}{\sqrt{558}} \underline{j} - \frac{10}{\sqrt{558}} \underline{k}$