7.2 Q-5

Question Statement

Find the unit vector in the direction of the given vector: $\underline{\mathbf{V}} = \underline{\mathbf{i}} + 2 \underline{\mathbf{j}} - \underline{\mathbf{k}}$

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Background and Explanation

A unit vector is a vector that has a magnitude of 1 but points in the same direction as the given vector. To find the unit vector in the direction of any vector $\mathbf{V}$, we use the following formula: $\hat{V} = \frac{\mathbf{V}}{|\mathbf{V}|}$

Where:

• $\hat{V}$ is the unit vector in the direction of $\mathbf{V}$.
• $|\mathbf{V}|$ is the magnitude of vector $\mathbf{V}$.

To find the magnitude of a vector $\mathbf{V} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k}$, we use: $|\mathbf{V}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

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Solution

We are given the vector: $\underline{\mathbf{V}} = \underline{\mathbf{i}} + 2 \underline{\mathbf{j}} - \underline{\mathbf{k}}$

Step 1: Find the magnitude of the vector

The magnitude of a vector $\mathbf{V} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k}$ is: $|\mathbf{V}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

For our vector:

• $v_x = 1$ (coefficient of $\mathbf{i}$),
• $v_y = 2$ (coefficient of $\mathbf{j}$),
• $v_z = -1$ (coefficient of $\mathbf{k}$).

So, the magnitude is: $|\mathbf{V}| = \sqrt{1^2 + 2^2 + (-1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}$

Step 2: Find the unit vector

To find the unit vector, we divide each component of the vector $\mathbf{V}$ by its magnitude $|\mathbf{V}|$. This gives the unit vector $\hat{V}$ in the direction of $\mathbf{V}$: $\hat{V} = \frac{\mathbf{V}}{|\mathbf{V}|}$

Substituting in the values: $\hat{V} = \frac{1}{\sqrt{6}} \left( \underline{\mathbf{i}} + 2 \underline{\mathbf{j}} - \underline{\mathbf{k}} \right)$

Thus, the unit vector is: $\hat{V} = \frac{1}{\sqrt{6}} \underline{\mathbf{i}} + \frac{2}{\sqrt{6}} \underline{\mathbf{j}} - \frac{1}{\sqrt{6}} \underline{\mathbf{k}}$

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Key Formulas or Methods Used

• Magnitude of a vector: $|\mathbf{V}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

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

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Summary of Steps

1. Calculate the magnitude of the given vector $\mathbf{V}$ using the formula $|\mathbf{V}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$.
2. Find the unit vector by dividing each component of $\mathbf{V}$ by its magnitude.
3. The resulting unit vector in the direction of $\mathbf{V}$ is: $\hat{V} = \frac{1}{\sqrt{6}} \underline{\mathbf{i}} + \frac{2}{\sqrt{6}} \underline{\mathbf{j}} - \frac{1}{\sqrt{6}} \underline{\mathbf{k}}$