Question

If the vectors $\vec{A}=a_{x}\hat{i}+a_{y}\hat{j}+a_{z}\hat{k}, \vec{B}=b_{x}\hat{i}+b_{y}\hat{j}+b_{z}\hat{k}$ and $\vec{C}=c_{x}\hat{i}+c_{y}\hat{j}+c_{z}\hat{k}$ are defining the vector $\vec{T}=\vec{A}-\vec{B}+\vec{C}$ then the z-component of the vector $T_{z}$ is

• A. $a_{x}-b_{x}+c_{z}$
• B. $a_{z}-b_{z}+c_{z}$
• C. $a_{y}-b_{z}+c_{z}$
• D. $a_{z}-b_{z}+c_{x}$
• E. $a_{y}-b_{y}+c_{y}$

Hint:

Logic Tip: In linear vector operations, independent axes never mix. You don't need to write out the full equation. The operation is $\vec{A}-\vec{B}+\vec{C}$. For the z-component, simply grab the z-terms and apply the same signs: $a_z - b_z + c_z$.

Answer 1

The Correct Option is B

Concept:
Vector addition and subtraction are performed component by component. To find a specific component of a resultant vector, you apply the given operations strictly to that corresponding axis component of the constituent vectors.
Step 1: Set up the vector operation.
We are given the operation $\vec{T} = \vec{A} - \vec{B} + \vec{C}$. Substitute the full vector expressions into the equation: $$\vec{T} = (a_x\hat{i} + a_y\hat{j} + a_z\hat{k}) - (b_x\hat{i} + b_y\hat{j} + b_z\hat{k}) + (c_x\hat{i} + c_y\hat{j} + c_z\hat{k})$$
Step 2: Group the terms by their unit vectors.
Combine all $\hat{i}$ components, $\hat{j}$ components, and $\hat{k}$ components together: $$\vec{T} = (a_x - b_x + c_x)\hat{i} + (a_y - b_y + c_y)\hat{j} + (a_z - b_z + c_z)\hat{k}$$
Step 3: Extract the required z-component.
The resultant vector is in the form $\vec{T} = T_x\hat{i} + T_y\hat{j} + T_z\hat{k}$. By direct comparison, the coefficient of the $\hat{k}$ unit vector is the z-component $T_z$. $$T_z = a_z - b_z + c_z$$