The projection of the vector `vecA = hati - 2hatj + hatk` on the vector `vecB = 4hati - 4hatj + 7hatk` is

Options

A. `(19)/(9)`
B. `(38)/(9)`
C. `(8)/(9)`
D. `(4)/(9)`

Explanation

To find the projection of the vector \(\vec{A} = \hat{i} - 2\hat{j} + \hat{k}\) on the vector \(\vec{B} = 4\hat{i} - 4\hat{j} + 7\hat{k}\), we can use the formula for the projection of one vector onto another. The projection of vector \(\vec{A}\) onto vector \(\vec{B}\) is given by:

\[ \text{Projection of } \vec{A} \text{ on } \vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} \]

### Step 1: Calculate the dot product \(\vec{A} \cdot \vec{B}\)

The dot product of two vectors \(\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) is calculated as:

\[ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3 \]

For our vectors:

- \(a_1 = 1\), \(a_2 = -2\), \(a_3 = 1\) (from \(\vec{A}\))
- \(b_1 = 4\), \(b_2 = -4\), \(b_3 = 7\) (from \(\vec{B}\))

Now, substituting these values into the dot product formula:

\[ \vec{A} \cdot \vec{B} = (1)(4) + (-2)(-4) + (1)(7) \]

Calculating each term:

\[ = 4 + 8 + 7 = 19 \]

### Step 2: Calculate the magnitude of vector \(\vec{B}\)

The magnitude of a vector \(\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) is given by:

\[ |\vec{B}| = \sqrt{b_1^2 + b_2^2 + b_3^2} \]

Substituting the values from \(\vec{B}\):

\[ |\vec{B}| = \sqrt{4^2 + (-4)^2 + 7^2} \]

Calculating each term:

\[ = \sqrt{16 + 16 + 49} = \sqrt{81} = 9 \]

### Step 3: Calculate the projection of \(\vec{A}\) on \(\vec{B}\)

Now, substituting the values of the dot product and the magnitude into the projection formula:

\[ \text{Projection of } \vec{A} \text{ on } \vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} = \frac{19}{9} \]

### Final Answer

The projection of vector \(\vec{A}\) on vector \(\vec{B}\) is:

\[ \frac{19}{9} \]

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