7.2 Q-3

Question Statement

Given the following vectors, find the magnitude of each vector and write their direction cosines:

(i) $\underline{\mathbf{V}} = 2 \underline{\mathbf{i}} + 3 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}$ (ii) $\underline{\mathbf{V}} = \underline{\mathbf{i}} - \underline{\mathbf{j}} - \underline{k}$ (iii) $\underline{\mathbf{V}} = 4 \underline{\mathbf{i}} - 5 \underline{\mathbf{j}} + 0 \underline{\mathbf{k}}$

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Background and Explanation

To solve this problem, you need to know the following concepts:

• Magnitude of a Vector: The magnitude of a vector $\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k}$ is calculated as: $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

• Direction Cosines: The direction cosines of a vector $\mathbf{v} = (v_x, v_y, v_z)$ are the cosines of the angles that the vector makes with the $x$-, $y$-, and $z$-axes. They are given by: $\cos \alpha = \frac{v_x}{|\mathbf{v}|}, \quad \cos \beta = \frac{v_y}{|\mathbf{v}|}, \quad \cos \gamma = \frac{v_z}{|\mathbf{v}|}$

These two formulas are key to finding both the magnitude and the direction cosines of any given vector.

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Solution

(i) Find the magnitude and direction cosines of $\underline{\mathbf{V}} = 2 \underline{\mathbf{i}} + 3 \underline{\mathbf{j}} + 4 \underline{\mathbf{k}}$

1. Magnitude: The magnitude of $\underline{\mathbf{V}}$ is calculated using the formula for the magnitude of a vector:

$$|\underline{\mathbf{V}}| = \sqrt{(2)^2 + (3)^2 + (4)^2}$$

Calculate each squared term:

$$|\underline{\mathbf{V}}| = \sqrt{4 + 9 + 16} = \sqrt{29}$$

So, the magnitude of $\underline{\mathbf{V}}$ is:

$$|\underline{\mathbf{V}}| = \sqrt{29}$$

2. Direction Cosines: The direction cosines of $\underline{\mathbf{V}}$ are calculated by dividing each component of the vector by its magnitude:

$$\cos \alpha = \frac{2}{\sqrt{29}}, \quad \cos \beta = \frac{3}{\sqrt{29}}, \quad \cos \gamma = \frac{4}{\sqrt{29}}$$

Therefore, the direction cosines of $\underline{\mathbf{V}}$ are:

$$\left[\frac{2}{\sqrt{29}}, \frac{3}{\sqrt{29}}, \frac{4}{\sqrt{29}}\right]$$

(ii) Find the magnitude and direction cosines of $\underline{\mathbf{V}} = \underline{\mathbf{i}} - \underline{\mathbf{j}} - \underline{k}$

1. Magnitude: The magnitude of $\underline{\mathbf{V}}$ is:

$$|\underline{\mathbf{V}}| = \sqrt{(1)^2 + (-1)^2 + (-1)^2}$$

Calculate each squared term:

$$|\underline{\mathbf{V}}| = \sqrt{1 + 1 + 1} = \sqrt{3}$$

So, the magnitude of $\underline{\mathbf{V}}$ is:

$$|\underline{\mathbf{V}}| = \sqrt{3}$$

2. Direction Cosines: The direction cosines of $\underline{\mathbf{V}}$ are:

$$\cos \alpha = \frac{1}{\sqrt{3}}, \quad \cos \beta = \frac{1}{\sqrt{3}}, \quad \cos \gamma = \frac{1}{\sqrt{3}}$$

Therefore, the direction cosines of $\underline{\mathbf{V}}$ are:

$$\left[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right]$$

(iii) Find the magnitude and direction cosines of $\underline{\mathbf{V}} = 4 \underline{\mathbf{i}} - 5 \underline{\mathbf{j}} + 0 \underline{\mathbf{k}}$

1. Magnitude: The magnitude of $\underline{\mathbf{V}}$ is:

$$|\underline{\mathbf{V}}| = \sqrt{(4)^2 + (-5)^2 + (0)^2}$$

Calculate each squared term:

$$|\underline{\mathbf{V}}| = \sqrt{16 + 25 + 0} = \sqrt{41}$$

So, the magnitude of $\underline{\mathbf{V}}$ is:

$$|\underline{\mathbf{V}}| = \sqrt{41}$$

2. Direction Cosines: The direction cosines of $\underline{\mathbf{V}}$ are:

$$\cos \alpha = \frac{4}{\sqrt{41}}, \quad \cos \beta = \frac{-5}{\sqrt{41}}, \quad \cos \gamma = \frac{0}{\sqrt{41}} = 0$$

Therefore, the direction cosines of $\underline{\mathbf{V}}$ are:

$$\left[\frac{4}{\sqrt{41}}, \frac{-5}{\sqrt{41}}, 0\right]$$

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Key Formulas or Methods Used

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

• Direction Cosines: $\cos \alpha = \frac{v_x}{|\mathbf{v}|}, \quad \cos \beta = \frac{v_y}{|\mathbf{v}|}, \quad \cos \gamma = \frac{v_z}{|\mathbf{v}|}$

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Summary of Steps

1. (i) Find the magnitude of $\underline{\mathbf{V}}$ using the formula $|\underline{\mathbf{V}}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$, then compute the direction cosines by dividing each component by the magnitude.
2. (ii) Repeat for $\underline{\mathbf{V}} = \underline{\mathbf{i}} - \underline{\mathbf{j}} - \underline{k}$.
3. (iii) Similarly, find the magnitude and direction cosines for $\underline{\mathbf{V}} = 4 \underline{\mathbf{i}} - 5 \underline{\mathbf{j}} + 0 \underline{\mathbf{k}}$.