If `vecb ne 0`, then every vector `veca` can be written in a unique manner as the sum of a vector `veca_(p)` parallel to `vecb` and a vector `veca_(q)` perpendicular to `vecb`. If `veca` is parallel to `vecb` then `veca_(q)`=0 and `veca_(q)=veca`. The vector `veca_(p)` is called the projection of `veca` on `vecb` and is denoted by proj `vecb(veca)`. Since proj`vecb(veca)` is parallel to `vecb`, it is a scalar multiple of the vector in the direction of `vecb` i.e.,
proj `vecb(veca)=lambdavecUvecb" " (vecUvecb=(vecb)/(|vecb|))`
The scalar `lambda` is called the componennt of `veca` in the direction of `vecb` and is denoted by comp `vecb(veca)`. In fact proj `vecb(veca)=(veca.vecUvecb)vecUvecb` and comp `vecb(veca)=veca.vecUvecb`.
If `veca=-2hati+hatj+hatk` and `vecb=4hati-3hatj+hatk` then proj `vecb(veca)` is

Options

A. a. `4hati+2hatj+3hatk`
B. b. `(-5)/(13)` `4hati-3hatj+hatk`
C. c. `2hatj+hatj+hatk`
D. d. `(-4)/(11)` `4hati-3hatj+hatk`

Answer

To solve the problem of finding the projection of vector \(\vec{a}\) onto vector \(\vec{b}\), we will follow these steps:

### Step 1: Identify the vectors
Given:
\[ \vec{a} = -2\hat{i} + \hat{j} + \hat{k} \]
\[ \vec{b} = 4\hat{i} - 3\hat{j} + \hat{k} \]

### Step 2: Calculate the dot product \(\vec{a} \cdot \vec{b}\)
The dot product is calculated as follows:
\[ \vec{a} \cdot \vec{b} = (-2)(4) + (1)(-3) + (1)(1) \]
\[ = -8 - 3 + 1 = -10 \]

### Step 3: Calculate the magnitude of \(\vec{b}\)
The magnitude of vector \(\vec{b}\) is given by:
\[ |\vec{b}| = \sqrt{(4)^2 + (-3)^2 + (1)^2} = \sqrt{16 + 9 + 1} = \sqrt{26} \]

### Step 4: Calculate the projection of \(\vec{a}\) onto \(\vec{b}\)
The formula for the projection of \(\vec{a}\) onto \(\vec{b}\) is:
\[ \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]
First, we need \( |\vec{b}|^2 \):
\[ |\vec{b}|^2 = 26 \]
Now substituting the values:
\[ \text{proj}_{\vec{b}} \vec{a} = \frac{-10}{26} \vec{b} = \frac{-5}{13} \vec{b} \]

### Step 5: Substitute \(\vec{b}\) into the projection formula
Substituting \(\vec{b}\) into the equation:
\[ \text{proj}_{\vec{b}} \vec{a} = \frac{-5}{13} (4\hat{i} - 3\hat{j} + \hat{k}) \]
Distributing the scalar:
\[ = \frac{-20}{13} \hat{i} + \frac{15}{13} \hat{j} - \frac{5}{13} \hat{k} \]

### Final Answer
Thus, the projection of \(\vec{a}\) onto \(\vec{b}\) is:
\[ \text{proj}_{\vec{b}} \vec{a} = \frac{-20}{13} \hat{i} + \frac{15}{13} \hat{j} - \frac{5}{13} \hat{k} \]

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