Question

If the liner \(\overrightarrow{r}=2\hat{i}+\hat{j}+t(3\hat{i}+\hat{j}-2\hat{k})\) is parallel to the plane 2x+4y+az=8, then the value of a is equal to

• A. 2
• B. 3
• C. 4
• D. 5
• E. 6

Answer 1

The Correct Option is D

The given equation of the line is: 

\[ \overrightarrow{r} = 2\hat{i} + \hat{j} + t(3\hat{i} + \hat{j} - 2\hat{k}) \]

Direction vector of the line:

\[ \overrightarrow{d} = 3\hat{i} + \hat{j} - 2\hat{k} \]

The given plane equation is:

\[ 2x + 4y + az = 8 \]

Normal vector of the plane:

\[ \overrightarrow{N} = 2\hat{i} + 4\hat{j} + a\hat{k} \]

Condition for parallelism: The line is parallel to the plane if the direction vector \( \overrightarrow{d} \) is perpendicular to the normal vector \( \overrightarrow{N} \), i.e.,

\[ \overrightarrow{d} \cdot \overrightarrow{N} = 0 \]

Computing the dot product:

\[ (3\hat{i} + \hat{j} - 2\hat{k}) \cdot (2\hat{i} + 4\hat{j} + a\hat{k}) \] \[ = (3 \times 2) + (1 \times 4) + (-2 \times a) \] \[ = 6 + 4 - 2a \]

Setting it to zero:

\[ 6 + 4 - 2a = 0 \]

Solving for \( a \):

\[ 10 = 2a \] \[ a = 5 \]

Final Answer:

\[ \mathbf{5} \]