Question

If $\vec{a} = (x+2y-3)\hat{i} + (2x-y+3)\hat{j}$ and $\vec{b} = (3x-2y)\hat{i} + (x-y+1)\hat{j}$ are two vectors such that $\vec{a} = 2\vec{b}$, then $y-5x=$

• A. 10
• B. -10
• C. 8
• D. -8

Hint:

When solving systems of linear equations derived from vector components, be very careful with algebraic manipulations. If your final result doesn't match the options, double-check your work. If it's still inconsistent, consider simple typos in the question, such as a sign flip or transposed variables in the final expression (e.g., $y-5x$ vs $5x-y$).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept: When two vectors are equal, their corresponding scalar components (coefficients of \( \bar{i} \) and \( \bar{j} \)) must be equal. Here, we are given \( \bar{a} = 2\bar{b} \).
Step 2: Setting up Equations: Comparing the components of \( \bar{i} \): \[ x + 2y - 3 = 2(3x - 2y) \] \[ x + 2y - 3 = 6x - 4y \] \[ 5x - 6y = -3 \quad \dots (1) \] Comparing the components of \( \bar{j} \): \[ 2x - y + 3 = 2(x - y + 1) \] \[ 2x - y + 3 = 2x - 2y + 2 \] Subtract \( 2x \) from both sides: \[ -y + 3 = -2y + 2 \] \[ 2y - y = 2 - 3 \] \[ y = -1 \]
Step 3: Solving for \( x \): Substitute \( y = -1 \) into equation (1): \[ 5x - 6(-1) = -3 \] \[ 5x + 6 = -3 \] \[ 5x = -9 \] \[ x = -\frac{9}{5} \]
Step 4: Finding the Target Value: We need to calculate \( y - 5x \): \[ y - 5x = -1 - 5\left(-\frac{9}{5}\right) \] \[ y - 5x = -1 - (-9) \] \[ y - 5x = -1 + 9 = 8 \]