Given the three vectors `vec(a) = - 2hati + hat(j) + hat(k), vec(b) = hat(i) + 5hat(j)` and `vec(c) = 4hat(i) + 4hat(j) - 2hat(k)`. The projection of the vector `3vec(a) - 2vec(b)` on the vector `vec(c)` is

Options

A. 11
B. `-11`
C. 13
D. none of these

Answer

To find the projection of the vector \(3\vec{a} - 2\vec{b}\) on the vector \(\vec{c}\), we will follow these steps:

### Step 1: Define the vectors
Given:
\[ \vec{a} = -2\hat{i} + \hat{j} + \hat{k} \]
\[ \vec{b} = \hat{i} + 5\hat{j} \]
\[ \vec{c} = 4\hat{i} + 4\hat{j} - 2\hat{k} \]

### Step 2: Calculate \(3\vec{a}\)
To find \(3\vec{a}\):
\[ 3\vec{a} = 3(-2\hat{i} + \hat{j} + \hat{k}) = -6\hat{i} + 3\hat{j} + 3\hat{k} \]

### Step 3: Calculate \(-2\vec{b}\)
To find \(-2\vec{b}\):
\[ -2\vec{b} = -2(\hat{i} + 5\hat{j}) = -2\hat{i} - 10\hat{j} \]

### Step 4: Calculate \(3\vec{a} - 2\vec{b}\)
Now, we will combine \(3\vec{a}\) and \(-2\vec{b}\):
\[ 3\vec{a} - 2\vec{b} = (-6\hat{i} + 3\hat{j} + 3\hat{k}) + (-2\hat{i} - 10\hat{j}) \]
\[ = (-6 - 2)\hat{i} + (3 - 10)\hat{j} + 3\hat{k} \]
\[ = -8\hat{i} - 7\hat{j} + 3\hat{k} \]

### Step 5: Calculate the projection of \(3\vec{a} - 2\vec{b}\) on \(\vec{c}\)
The formula for the projection of vector \(\vec{u}\) on vector \(\vec{v}\) is given by:
\[ \text{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2} \vec{v} \]
In our case, \(\vec{u} = 3\vec{a} - 2\vec{b}\) and \(\vec{v} = \vec{c}\).

### Step 6: Calculate the dot product \((3\vec{a} - 2\vec{b}) \cdot \vec{c}\)
\[ \vec{u} \cdot \vec{c} = (-8\hat{i} - 7\hat{j} + 3\hat{k}) \cdot (4\hat{i} + 4\hat{j} - 2\hat{k}) \]
\[ = (-8)(4) + (-7)(4) + (3)(-2) \]
\[ = -32 - 28 - 6 = -66 \]

### Step 7: Calculate the magnitude of \(\vec{c}\)
\[ |\vec{c}| = \sqrt{(4)^2 + (4)^2 + (-2)^2} = \sqrt{16 + 16 + 4} = \sqrt{36} = 6 \]

### Step 8: Calculate the projection
Now we can find the projection:
\[ \text{proj}_{\vec{c}}(3\vec{a} - 2\vec{b}) = \frac{-66}{6^2} \vec{c} = \frac{-66}{36} \vec{c} = \frac{-11}{6} \vec{c} \]

### Final Answer
The projection of the vector \(3\vec{a} - 2\vec{b}\) on the vector \(\vec{c}\) is:
\[ \frac{-11}{6} \vec{c} \]