Question

If \( \vec{f} = i + j + k \) and \( \vec{g} = 2i - j + 3k \), then the projection vector of \( \vec{f} \) on \( \vec{g} \) is:

• A. \( \frac{2}{7} (i + j + k) \)
• B. \( \frac{2}{7} (2i - j + 3k) \)
• C. \( \frac{1}{3} (i + j + k) \)
• D. \( \frac{1}{14} (2i - j + 3k) \)

Hint:

The projection formula \( \text{Proj}_{\vec{g}} (\vec{f}) = \frac{\vec{f} \cdot \vec{g}}{|\vec{g}|^2} \vec{g} \) is useful in many vector calculations, including physics and engineering.

Answer 1

The Correct Option is B

To find the projection vector of \(\vec{f}\) on \(\vec{g}\), we use the formula for the projection of vector \(\vec{a}\) onto vector \(\vec{b}\):

\(\text{proj}_{\vec{b}}\vec{a}=\left(\frac{\vec{a}\cdot\vec{b}}{\vec{b}\cdot\vec{b}}\right)\vec{b}\)

Given \(\vec{f}=i+j+k\) and \(\vec{g}=2i-j+3k\), we first calculate the dot product \(\vec{f}\cdot\vec{g}\):

\(\vec{f}\cdot\vec{g}=(i+j+k)\cdot(2i-j+3k)=1\cdot2+1\cdot(-1)+1\cdot3=2-1+3=4\)

Next, compute \(\vec{g}\cdot\vec{g}\):

\(\vec{g}\cdot\vec{g}=(2i-j+3k)\cdot(2i-j+3k)=2^2+(-1)^2+3^2=4+1+9=14\)

Now substitute these into the projection formula:

\(\text{proj}_{\vec{g}}\vec{f}=\left(\frac{4}{14}\right)(2i-j+3k)=\frac{2}{7}(2i-j+3k)\)

Thus, the projection vector of \(\vec{f}\) on \(\vec{g}\) is \(\frac{2}{7}(2i-j+3k)\).

Answer 2

We are given two vectors: \[ \vec{f} = \hat{i} + \hat{j} + \hat{k}, \quad \vec{g} = 2\hat{i} - \hat{j} + 3\hat{k} \] Step 1: Recall the Formula for Projection The projection vector of \( \vec{f} \) on \( \vec{g} \) is given by: \[ \text{Proj}_{\vec{g}} \vec{f} = \frac{\vec{f} \cdot \vec{g}}{|\vec{g}|^2} \vec{g} \] Step 2: Compute the Dot Product \[ \vec{f} \cdot \vec{g} = (1)(2) + (1)(-1) + (1)(3) \] \[ \vec{f} \cdot \vec{g} = 2 - 1 + 3 = 4 \] Step 3: Compute \( |\vec{g}|^2 \) \[ |\vec{g}|^2 = (2)^2 + (-1)^2 + (3)^2 \] \[ |\vec{g}|^2 = 4 + 1 + 9 = 14 \] Step 4: Compute the Projection Vector \[ \text{Proj}_{\vec{g}} \vec{f} = \frac{4}{14} \vec{g} = \frac{2}{7} \vec{g} \] \[ = \frac{2}{7} (2\hat{i} - \hat{j} + 3\hat{k}) \] Step 5: Final Answer 

\[Correct Answer: (2) \ \frac{2}{7} (2\hat{i} - \hat{j} + 3\hat{k})\]