If `vec(a) = 2hat(i) + hat(j) + 2hat(k), vec(b) = 5hat(i) - 3hat(j) + hat(k)`, then projection of `vec(a)` on `vec(b)` is

Options

A. `3hat(i) - 3hat(j) + hat(k)`
B. `(9(5hat(i) - 3hat(j) + hat(k))/(7))`
C. `((5hat(i) - 3hat(j) - hat(k)))/(35)`
D. `9(5hat(i) - 3hat(j) + hat(k))`

Answer

To find the projection of vector **a** on vector **b**, we can use the formula for the projection of one vector onto another:

\[ \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]

Where:
- \(\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}\)
- \(\vec{b} = 5\hat{i} - 3\hat{j} + \hat{k}\)

### 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:

\[ \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 \]

Substituting the values:

\[ \vec{a} \cdot \vec{b} = (2)(5) + (1)(-3) + (2)(1) \]

Calculating this:

\[ = 10 - 3 + 2 = 9 \]

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

The magnitude of vector \(\vec{b}\) is given by:

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

Substituting the values:

\[ |\vec{b}| = \sqrt{5^2 + (-3)^2 + 1^2} \]

Calculating this:

\[ = \sqrt{25 + 9 + 1} = \sqrt{35} \]

### Step 3: Calculate \(|\vec{b}|^2\)

Now, we need \(|\vec{b}|^2\):

\[ |\vec{b}|^2 = 35 \]

### Step 4: Substitute into the projection formula

Now we can substitute the values into the projection formula:

\[ \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} = \frac{9}{35} \vec{b} \]

Substituting \(\vec{b}\):

\[ = \frac{9}{35} (5\hat{i} - 3\hat{j} + \hat{k}) \]

### Step 5: Distributing the scalar

Distributing \(\frac{9}{35}\):

\[ = \left(\frac{9 \cdot 5}{35}\right) \hat{i} + \left(\frac{9 \cdot (-3)}{35}\right) \hat{j} + \left(\frac{9 \cdot 1}{35}\right) \hat{k} \]

Calculating each component:

\[ = \frac{45}{35} \hat{i} - \frac{27}{35} \hat{j} + \frac{9}{35} \hat{k} \]

### Final Result

Thus, the projection of vector **a** on vector **b** is:

\[ \text{proj}_{\vec{b}} \vec{a} = \frac{9}{35} \hat{i} - \frac{27}{35} \hat{j} + \frac{9}{35} \hat{k} \]

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