Let `veca= 2 hati + 3hatj - 6hatk, vecb = 2hati - 3hatj + 6hatk and vecc = -2 hati + 3hatj + 6hatk`. Let `veca_(1)` be the projection of `veca` on `vecb and veca_(2)` be the projection of `veca_(1)` on `vecc` . Then
`veca_(2)` is equal to

Options

A. (a) `943/49 (2 hati - 3hatj - 6hatk)`
B. (b) `943/(49^(2)) (2 hati - 3hatj - 6hatk)`
C. (c) `943/49 (-2 hati + 3hatj + 6hatk)`
D. (d) `943/(49^(2)) (-2 hati + 3hatj + 6hatk)`

Explanation

To solve the problem step by step, we will find the projection of vector \(\vec{a}\) onto vector \(\vec{b}\) to get \(\vec{a_1}\), and then project \(\vec{a_1}\) onto vector \(\vec{c}\) to find \(\vec{a_2}\).

### Step 1: Define the vectors
Given:
\[ \vec{a} = 2\hat{i} + 3\hat{j} - 6\hat{k} \]
\[ \vec{b} = 2\hat{i} - 3\hat{j} + 6\hat{k} \]
\[ \vec{c} = -2\hat{i} + 3\hat{j} + 6\hat{k} \]

### Step 2: Calculate the dot product \(\vec{a} \cdot \vec{b}\)
\[ \vec{a} \cdot \vec{b} = (2)(2) + (3)(-3) + (-6)(6) \]
\[ = 4 - 9 - 36 = -41 \]

### Step 3: Calculate the magnitude of \(\vec{b}\)
\[ |\vec{b}|^2 = (2)^2 + (-3)^2 + (6)^2 = 4 + 9 + 36 = 49 \]

### Step 4: Calculate the projection \(\vec{a_1}\) of \(\vec{a}\) onto \(\vec{b}\)
The formula for projection is:
\[ \vec{a_1} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]
Substituting the values:
\[ \vec{a_1} = \frac{-41}{49} \vec{b} \]
\[ = \frac{-41}{49} (2\hat{i} - 3\hat{j} + 6\hat{k}) = \left(\frac{-82}{49}\hat{i} + \frac{123}{49}\hat{j} - \frac{246}{49}\hat{k}\right) \]

### Step 5: Calculate the dot product \(\vec{a_1} \cdot \vec{c}\)
\[ \vec{a_1} \cdot \vec{c} = \left(\frac{-82}{49}\right)(-2) + \left(\frac{123}{49}\right)(3) + \left(\frac{-246}{49}\right)(6) \]
\[ = \frac{164}{49} + \frac{369}{49} - \frac{1476}{49} \]
\[ = \frac{164 + 369 - 1476}{49} = \frac{-943}{49} \]

### Step 6: Calculate the magnitude of \(\vec{c}\)
\[ |\vec{c}|^2 = (-2)^2 + (3)^2 + (6)^2 = 4 + 9 + 36 = 49 \]

### Step 7: Calculate the projection \(\vec{a_2}\) of \(\vec{a_1}\) onto \(\vec{c}\)
Using the projection formula:
\[ \vec{a_2} = \frac{\vec{a_1} \cdot \vec{c}}{|\vec{c}|^2} \vec{c} \]
Substituting the values:
\[ \vec{a_2} = \frac{-943/49}{49} \vec{c} = \frac{-943}{2401} \vec{c} \]
\[ = \frac{-943}{2401} (-2\hat{i} + 3\hat{j} + 6\hat{k}) = \left(\frac{1886}{2401}\hat{i} - \frac{2829}{2401}\hat{j} - \frac{5658}{2401}\hat{k}\right) \]

### Final Result
Thus, the projection \(\vec{a_2}\) is:
\[ \vec{a_2} = \frac{943}{2401} (2\hat{i} - 3\hat{j} - 6\hat{k}) \]