7.2 Q-7

Question Statement

Find the vectors that satisfy the following conditions:

1. (i) A vector whose magnitude is 4 and is parallel to $2 \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 6 \underline{\mathbf{k}}$.
2. (ii) A vector whose magnitude is 2 and is parallel to $-\underline{\mathbf{i}} + \underline{\mathbf{j}} + \underline{\mathbf{k}}$.

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Background and Explanation

To solve these problems, we need to apply two key concepts:

1. 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: $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$

2. Unit Vector: A unit vector has a magnitude of 1 and points in the same direction as the original vector. To find a unit vector $\hat{v}$ in the direction of a vector $\mathbf{v}$, we divide each component of $\mathbf{v}$ by its magnitude: $\hat{v} = \frac{\mathbf{v}}{|\mathbf{v}|}$

Once we have the unit vector, we can easily scale it to obtain the required magnitude.

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Solution

(i) Find a vector whose magnitude is 4 and is parallel to $2 \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 6 \underline{\mathbf{k}}$

1. Find the magnitude of the given vector: The given vector is $2 \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 6 \underline{\mathbf{k}}$. Its magnitude is:

$$|\mathbf{V}| = \sqrt{(2)^2 + (-3)^2 + (6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7$$

2. Find the unit vector in the direction of the given vector: The unit vector $\hat{V}$ in the direction of the vector $\mathbf{V}$ is obtained by dividing each component of $\mathbf{V}$ by its magnitude:

$$\widehat{V} = \frac{\mathbf{V}}{|\mathbf{V}|} = \frac{2 \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 6 \underline{\mathbf{k}}}{7}$$

3. Scale the unit vector to obtain a vector of magnitude 4: The required vector is obtained by multiplying the unit vector by 4:

$$4 \hat{V} = \frac{4 \left( 2 \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 6 \underline{\mathbf{k}} \right)}{7} = \frac{8}{7} \underline{\mathbf{i}} - \frac{12}{7} \underline{\mathbf{j}} - \frac{24}{7} \underline{\mathbf{k}}$$

Thus, the required vector is: $\frac{8}{7} \underline{\mathbf{i}} - \frac{12}{7} \underline{\mathbf{j}} - \frac{24}{7} \underline{\mathbf{k}}$

(ii) Find a vector whose magnitude is 2 and is parallel to $-\underline{\mathbf{i}} + \underline{\mathbf{j}} + \underline{\mathbf{k}}$

1. Find the magnitude of the given vector: The given vector is $-\underline{\mathbf{i}} + \underline{\mathbf{j}} + \underline{\mathbf{k}}$. Its magnitude is:

$$|\mathbf{V}| = \sqrt{(-1)^2 + (1)^2 + (1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3}$$

2. Find the unit vector in the direction of the given vector: The unit vector $\hat{V}$ in the direction of $\mathbf{V}$ is obtained by dividing each component of $\mathbf{V}$ by its magnitude:

$$\widehat{V} = \frac{\mathbf{V}}{|\mathbf{V}|} = \frac{-\underline{\mathbf{i}} + \underline{\mathbf{j}} + \underline{\mathbf{k}}}{\sqrt{3}}$$

3. Scale the unit vector to obtain a vector of magnitude 2: The required vector is obtained by multiplying the unit vector by 2:

$$2 \hat{V} = \frac{2 \left( -\underline{\mathbf{i}} + \underline{\mathbf{j}} + \underline{\mathbf{k}} \right)}{\sqrt{3}} = \frac{2}{\sqrt{3}} \underline{\mathbf{i}} + \frac{2}{\sqrt{3}} \underline{\mathbf{j}} + \frac{2}{\sqrt{3}} \underline{\mathbf{k}}$$

Thus, the required vector is: $\frac{2}{\sqrt{3}} \underline{\mathbf{i}} + \frac{2}{\sqrt{3}} \underline{\mathbf{j}} + \frac{2}{\sqrt{3}} \underline{\mathbf{k}}$

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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: $\hat{v} = \frac{\mathbf{v}}{|\mathbf{v}|}$

• Scaling a vector: To change the magnitude of a vector without changing its direction, multiply each component by the desired scalar.

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Summary of Steps

1. (i) Calculate the magnitude of the given vector $2 \underline{\mathbf{i}} - 3 \underline{\mathbf{j}} + 6 \underline{\mathbf{k}}$, find the unit vector, and scale it to obtain the required magnitude of 4.
2. (ii) Calculate the magnitude of the given vector $-\underline{\mathbf{i}} + \underline{\mathbf{j}} + \underline{\mathbf{k}}$, find the unit vector, and scale it to obtain the required magnitude of 2.
3. The required vectors are:
   • $\frac{8}{7} \underline{\mathbf{i}} - \frac{12}{7} \underline{\mathbf{j}} - \frac{24}{7} \underline{\mathbf{k}}$
   • $\frac{2}{\sqrt{3}} \underline{\mathbf{i}} + \frac{2}{\sqrt{3}} \underline{\mathbf{j}} + \frac{2}{\sqrt{3}} \underline{\mathbf{k}}$