A force `vecF = (hati - 8hatj - 7hatk)` is resolved along the mutually perpendicular directions , one of which is in the direction of `veca = 2hati + 2hatj + hatk` . Then the component of `vecF` in the direction of `vec a ` is

Options

A. ` - 14 hati - 14 hatj - 7hatk`
B. `-7/3 (2hati + 2hatj + hatk)`
C. `(-2hati - 2hatj -hatk)/(3)`
D. `7/3 (2hati + 2hatj + hatk)`

Answer

To find the component of the force vector \(\vec{F} = \hat{i} - 8\hat{j} - 7\hat{k}\) in the direction of the vector \(\vec{a} = 2\hat{i} + 2\hat{j} + \hat{k}\), we can follow these steps:

### Step 1: Find the unit vector in the direction of \(\vec{a}\)

The unit vector \(\hat{a}\) in the direction of \(\vec{a}\) is given by:

\[ \hat{a} = \frac{\vec{a}}{|\vec{a}|} \]

First, we need to calculate the magnitude of \(\vec{a}\):

\[ |\vec{a}| = \sqrt{(2)^2 + (2)^2 + (1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

Now, we can find the unit vector \(\hat{a}\):

\[ \hat{a} = \frac{2\hat{i} + 2\hat{j} + \hat{k}}{3} = \frac{2}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{1}{3}\hat{k} \]

### Step 2: Calculate the dot product \(\vec{F} \cdot \hat{a}\)

Next, we need to calculate the dot product of \(\vec{F}\) and \(\hat{a}\):

\[ \vec{F} \cdot \hat{a} = (1\hat{i} - 8\hat{j} - 7\hat{k}) \cdot \left(\frac{2}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{1}{3}\hat{k}\right) \]

Calculating the dot product:

\[ \vec{F} \cdot \hat{a} = 1 \cdot \frac{2}{3} + (-8) \cdot \frac{2}{3} + (-7) \cdot \frac{1}{3} \]

\[ = \frac{2}{3} - \frac{16}{3} - \frac{7}{3} = \frac{2 - 16 - 7}{3} = \frac{-21}{3} = -7 \]

### Step 3: Find the component of \(\vec{F}\) in the direction of \(\vec{a}\)

The component of \(\vec{F}\) in the direction of \(\vec{a}\) is given by:

\[ \text{Component of } \vec{F} \text{ along } \vec{a} = (\vec{F} \cdot \hat{a}) \hat{a} \]

Substituting the values:

\[ \text{Component of } \vec{F} \text{ along } \vec{a} = -7 \left(\frac{2}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{1}{3}\hat{k}\right) \]

\[ = -\frac{14}{3}\hat{i} - \frac{14}{3}\hat{j} - \frac{7}{3}\hat{k} \]

### Final Answer

Thus, the component of \(\vec{F}\) in the direction of \(\vec{a}\) is:

\[ -\frac{7}{3} \left(2\hat{i} + 2\hat{j} + \hat{k}\right) \]