If two vectors P = $\hat{i} $+2m$\hat{j}$+m$\hat{k}$ & Q = 4$\hat{i}$-2$\hat{j}$+m$\hat{k}$ are perpendicular to each other, then the value of m will be:

Question

Vector of magnitude 3 making equal angles with $x$ and $y$ axes and perpendicular to $z$ axis is

Answer 1

To determine the value of $ m $ for which the vectors $ \mathbf{P} = \hat{i} + 2m \hat{j} + m \hat{k} $ and $ \mathbf{Q} = 4\hat{i} - 2\hat{j} + m\hat{k} $ are perpendicular, we need to use the concept of the dot product.

The dot product of two vectors $ \mathbf{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} $ and $ \mathbf{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} $ is given by:

$\mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3$

Two vectors are perpendicular if their dot product is zero. Therefore, using the components of vectors $ \mathbf{P} $ and $ \mathbf{Q} $, we set up the equation:

$1 \times 4 + 2m \times (-2) + m \times m = 0$

Simplifying this equation gives:

$4 - 4m + m^2 = 0$

We can rearrange the equation as a quadratic in $ m $:

$m^2 - 4m + 4 = 0$

This can be rewritten as:

$ (m - 2)^2 = 0$

Solving this gives:

$m = 2$

Thus, the value of $ m $ that makes vectors $ \mathbf{P} $ and $ \mathbf{Q} $ perpendicular is $ m = 2 $.

Conclusion: The correct answer is $ m = 2 $.

Answer 2

To determine the value of $ m $ for which the vectors $ \mathbf{P} = \hat{i} + 2m \hat{j} + m \hat{k} $ and $ \mathbf{Q} = 4\hat{i} - 2\hat{j} + m\hat{k} $ are perpendicular, we need to use the concept of the dot product.

The dot product of two vectors $ \mathbf{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} $ and $ \mathbf{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} $ is given by:

$\mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3$

Two vectors are perpendicular if their dot product is zero. Therefore, using the components of vectors $ \mathbf{P} $ and $ \mathbf{Q} $, we set up the equation:

$1 \times 4 + 2m \times (-2) + m \times m = 0$

Simplifying this equation gives:

$4 - 4m + m^2 = 0$

We can rearrange the equation as a quadratic in $ m $:

$m^2 - 4m + 4 = 0$

This can be rewritten as:

$ (m - 2)^2 = 0$

Solving this gives:

$m = 2$

Thus, the value of $ m $ that makes vectors $ \mathbf{P} $ and $ \mathbf{Q} $ perpendicular is $ m = 2 $.

Conclusion: The correct answer is $ m = 2 $.

If two vectors P = \(\hat{i} \)+2m\(\hat{j}\)+m\(\hat{k}\) & Q = 4\(\hat{i}\)-2\(\hat{j}\)+m\(\hat{k}\) are perpendicular to each other, then the value of m will be:

Show Hint

The Correct Option is B

Solution and Explanation

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