NIMCET 2009 — Mathematics PYQ

1. Given Data:

• Vector $\vec{B}=3\hat{i}+0\hat{j}+4\hat{k}$

• Vector $\vec{A}=\hat{i}+\hat{j}+0\hat{k}$

• $\vec{B}={\vec{B}}_1+{\vec{B}}_2$

• ${\vec{B}}_1\parallel \vec{A}$ and ${\vec{B}}_2\perp \vec{A}$

2. Formula for Parallel Component (${\vec{B}}_1$):

The projection of $\vec{B}$ onto $\vec{A}$ is given by:

3. Step-by-Step Calculation:

• Calculate the dot product ($\vec{B}\cdot \vec{A}$):

  $$\vec{B}\cdot \vec{A}=(3)(1)+(0)(1)+(4)(0)$$
  $$\vec{B}\cdot \vec{A}=3+0+0=3$$

• Calculate the magnitude squared of $\vec{A}$ ($|\vec{A}{|}^2$):

  $$|\vec{A}{|}^2=(1{)}^2+(1{)}^2+(0{)}^2$$
  $$|\vec{A}{|}^2=1+1=2$$

• Find ${\vec{B}}_1$:

  $${\vec{B}}_1=\left(\frac{3}{2}\right)\vec{A}$$
  $${\vec{B}}_1=\frac{3}{2}(\hat{i}+\hat{j})$$

Final Answer:

The vector ${\vec{B}}_1$ is $\frac{3}{2}(\hat{i}+\hat{j})$. the correct option is (a).