Question:
If $2\hat{i} - \hat{j} + \hat{k} = s(3\hat{i} - 4\hat{j} - 4\hat{k}) + t(\hat{i} - 3\hat{j} - 5\hat{k})$, where $s$ and $t$ are scalars, then $3s + 5t$ is equal to:
If $2\hat{i} - \hat{j} + \hat{k} = s(3\hat{i} - 4\hat{j} - 4\hat{k}) + t(\hat{i} - 3\hat{j} - 5\hat{k})$, where $s$ and $t$ are scalars, then $3s + 5t$ is equal to:
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Solve vector equations by equating coefficients of $\hat{i}, \hat{j}, \hat{k}$.
Updated On: Apr 24, 2026
- $2$
- $-4$
- $-2$
- $6$
- $14$
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The Correct Option is C
Solution and Explanation
Concept:
• Equate coefficients of $\hat{i}, \hat{j}, \hat{k}$
Step 1: Expand RHS
\[ s(3,-4,-4) + t(1,-3,-5) = (3s+t, -4s-3t, -4s-5t) \]
Step 2: Equate components
\[ 3s + t = 2 \quad ...(1) \] \[ -4s - 3t = -1 \quad ...(2) \] \[ -4s - 5t = 1 \quad ...(3) \]
Step 3: Solve equations
From (2) and (3): \[ (-4s - 5t) - (-4s - 3t) = 1 - (-1) \] \[ -2t = 2 \Rightarrow t = -1 \] Substitute in (1): \[ 3s - 1 = 2 \Rightarrow 3s = 3 \Rightarrow s = 1 \]
Step 4: Find required value
\[ 3s + 5t = 3(1) + 5(-1) = 3 - 5 = -2 \] Final Conclusion:
\[ = -2 \]
• Equate coefficients of $\hat{i}, \hat{j}, \hat{k}$
Step 1: Expand RHS
\[ s(3,-4,-4) + t(1,-3,-5) = (3s+t, -4s-3t, -4s-5t) \]
Step 2: Equate components
\[ 3s + t = 2 \quad ...(1) \] \[ -4s - 3t = -1 \quad ...(2) \] \[ -4s - 5t = 1 \quad ...(3) \]
Step 3: Solve equations
From (2) and (3): \[ (-4s - 5t) - (-4s - 3t) = 1 - (-1) \] \[ -2t = 2 \Rightarrow t = -1 \] Substitute in (1): \[ 3s - 1 = 2 \Rightarrow 3s = 3 \Rightarrow s = 1 \]
Step 4: Find required value
\[ 3s + 5t = 3(1) + 5(-1) = 3 - 5 = -2 \] Final Conclusion:
\[ = -2 \]
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