Find the scalar projection of :
`vec(a)=7hat(i)+hat(j)-4hat(k)` on `vec(b)=2hat(i)+6hat(j)+3hat(k)`

Answer

To find the scalar projection of vector **a** on vector **b**, we can use the formula:

\[ \text{Scalar Projection of } \vec{a} \text{ on } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \]

Where:
- \(\vec{a} = 7\hat{i} + \hat{j} - 4\hat{k}\)
- \(\vec{b} = 2\hat{i} + 6\hat{j} + 3\hat{k}\)

### 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 given by:

\[ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \]

For our vectors:
- \(a_1 = 7\), \(a_2 = 1\), \(a_3 = -4\)
- \(b_1 = 2\), \(b_2 = 6\), \(b_3 = 3\)

Calculating the dot product:

\[ \vec{a} \cdot \vec{b} = (7)(2) + (1)(6) + (-4)(3) \]
\[ = 14 + 6 - 12 \]
\[ = 8 \]

### 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:

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

Calculating the magnitude of \(\vec{b}\):

\[ |\vec{b}| = \sqrt{(2)^2 + (6)^2 + (3)^2} \]
\[ = \sqrt{4 + 36 + 9} \]
\[ = \sqrt{49} \]
\[ = 7 \]

### Step 3: Calculate the scalar projection of \(\vec{a}\) on \(\vec{b}\)

Now we can substitute the values we found into the scalar projection formula:

\[ \text{Scalar Projection of } \vec{a} \text{ on } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \]
\[ = \frac{8}{7} \]

### Final Answer

The scalar projection of vector **a** on vector **b** is:

\[ \frac{8}{7} \]

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