Question

If $\vec{a} = \lambda x \hat{i} + y \hat{j} + 4z \hat{k}$, $\vec{b} = x \hat{i} + y \hat{j} + 3y \hat{k}$, and $\vec{c} = -2 \hat{i} - 2z \hat{j} - (\lambda + 1) \hat{k}$ such that $\vec{a} + \vec{b} - \vec{c} = \vec{0}$, then the value of $\lambda$ is ______.

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

Whenever vector equality creates a system of 3 equations with 4 variables, checking the provided multiple-choice options by plugging them directly into the simplest relationship is the fastest way to force a consistent state!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given an algebraic vector identity $\vec{a} + \vec{b} - \vec{c} = \vec{0}$. We must compare the corresponding $\hat{i}, \hat{j}$, and $\hat{k}$ components to solve for the unknown parameter $\lambda$.

Step 2: Detailed Explanation:
The given equation is $\vec{a} + \vec{b} - \vec{c} = \vec{0}$, which can be rearranged as:
$\vec{a} + \vec{b} = \vec{c}$
Let's sum vectors $\vec{a}$ and $\vec{b}$:
$\vec{a} + \vec{b} = (\lambda x + x)\hat{i} + (y + y)\hat{j} + (4z + 3y)\hat{k}$
$\vec{a} + \vec{b} = x(\lambda + 1)\hat{i} + 2y\hat{j} + (4z + 3y)\hat{k}$
We are given vector $\vec{c}$:
$\vec{c} = -2\hat{i} - 2z\hat{j} - (\lambda + 1)\hat{k}$
Since $\vec{a} + \vec{b} = \vec{c}$, we strictly equate the corresponding directional components:
1. Equating $\hat{j$ components:}
$2y = -2z \implies y = -z$ --- (Equation 1)
2. Equating $\hat{k$ components:}
$4z + 3y = -(\lambda + 1)$
Substitute $y = -z$ from Eq 1:
$4z + 3(-z) = -(\lambda + 1)$
$z = -(\lambda + 1)$ --- (Equation 2)
3. Equating $\hat{i$ components:}
$x(\lambda + 1) = -2$
From Equation 2, we know that $(\lambda + 1) = -z$. Substitute this:
$x(-z) = -2 \implies xz = 2$.
Assuming a standard consistent integer solution case intended by the framing (e.g. $x=1, z=2$ or $x=-1, z=-2$), let's test $\lambda = 1$ (Option b):
If $\lambda = 1$, then $(\lambda + 1) = 2$.
From Eq 2: $z = -2$.
From Eq 1: $y = 2$.
From $\hat{i}$ comp: $x(2) = -2 \implies x = -1$.
This forms a perfect, non-contradictory closed loop of values $(x=-1, y=2, z=-2)$ that mathematically satisfies every single constraint.

Step 3: Final Answer:
The value of $\lambda$ is 1, matching option (b).