Question

If $\vec{a} = 4\hat{i} + 13\hat{j} - 18\hat{k}$, $\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}$, and $\vec{c} = 2\hat{i} + 3\hat{j} - 4\hat{k}$ are three vectors such that $\vec{a} = x\vec{b} + y\vec{c}$, then $x + y = $

• A. $-1$
• B. $-2$
• C. $5$
• D. $1$

Hint:

Instead of solving equations completely, you can sometimes look for patterns in the coefficients. Adding equation (1) and equation (3) directly gives: $(x + 2y) + (3x - 4y) = 4 + (-18) \implies 4x - 2y = -14 \implies 2x - y = -7$. Adding this to $x + 2y = 4$ helps verify solutions rapidly!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks us to find the sum of scalars $x$ and $y$ such that the vector $\vec{a}$ is expressed as a linear combination of the vectors $\vec{b}$ and $\vec{c}$ via the relation $\vec{a} = x\vec{b} + y\vec{c}$.

Step 2: Key Formula or Approach:
We substitute the given component forms of vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ into the linear equation. By equating the corresponding coefficients of $\hat{i}$, $\hat{j}$, and $\hat{k}$ from both sides, we form a system of simultaneous linear equations to determine $x$ and $y$.

Step 3: Detailed Explanation:
Substitute the vectors into the relation $\vec{a} = x\vec{b} + y\vec{c}$: $$4\hat{i} + 13\hat{j} - 18\hat{k} = x(\hat{i} - 2\hat{j} + 3\hat{k}) + y(2\hat{i} + 3\hat{j} - 4\hat{k})$$ $$4\hat{i} + 13\hat{j} - 18\hat{k} = (x + 2y)\hat{i} + (-2x + 3y)\hat{j} + (3x - 4y)\hat{k}$$ Equating components on both sides gives three separate linear equations: align x + 2y &= 4
-2x + 3y &= 13
3x - 4y &= -18 align Let's solve equations (1) and (2). From equation (1), isolate $x$: $$x = 4 - 2y$$ Substitute this value into equation (2): $$-2(4 - 2y) + 3y = 13$$ $$-8 + 4y + 3y = 13$$ $$7y = 21 \implies y = 3$$ Substitute $y = 3$ back into the expression for $x$: $$x = 4 - 2(3) = 4 - 6 = -2$$ Let's verify these values using equation (3): $$\text{LHS} = 3(-2) - 4(3) = -6 - 12 = -18 = \text{RHS}$$ The values are consistent. Now calculate $x + y$: $$x + y = -2 + 3 = 1$$

Step 4: Final Answer:
The value of $x + y$ is $1$, which corresponds to option (D).