Vector Algebra · JEE Advanced previous year questions

Let $\vec{w}=\hat{ı}+\hat{ȷ}-2\hat{k}$, and $\vec{u}$ and $\vec{v}$ be two vectors, such that $\vec{u}\times \vec{v}=\vec{w}$ and $\vec{v}\times \vec{w}=\vec{u}$. Let $\alpha ,\beta ,\gamma$, and $t$ be real numbers such that $$\vec{u}=\alpha \hat{ı}+\beta \hat{ȷ}+\gamma \hat{k},-t\alpha +\beta +\gamma =0,\alpha -t\beta +\gamma =0,\text{and }\alpha +\beta -t\gamma =0.$$ Match each entry in List-I to the correct entry in List-II and choose the correct option. $$\begin{array}{ll}\text{List-I}& \text{List-II}\\ \text{(P) }|\vec{v}{|}^2\text{ is equal to}& \text{(1) }0\\ \text{(Q) If }\alpha =\sqrt{3}\text{, then }{\gamma }^2\text{ is equal to}& \text{(2) }1\\ \text{(R) If }\alpha =\sqrt{3}\text{, then }(\beta +\gamma {)}^2\text{ is equal to}& \text{(3) }2\\ \text{(S) If }\alpha =\sqrt{2}\text{, then }t+3\text{ is equal to}& \text{(4) }3\\ & \text{(5) }5\end{array}$$

1. A.(P) → (2) (Q) → (1) (R) → (4) (S) → (5)
2. B.(P) → (2) (Q) → (4) (R) → (3) (S) → (5)
3. C.(P) → (2) (Q) → (1) (R) → (4) (S) → (3)
4. D.(P) → (5) (Q) → (4) (R) → (1) (S) → (3)

Consider the vectors $$\vec{x}=\hat{ı}+2\hat{ȷ}+3\hat{k},\vec{y}=2\hat{ı}+3\hat{ȷ}+\hat{k},\text{and}\vec{z}=3\hat{ı}+\hat{ȷ}+2\hat{k}.$$ For two distinct positive real numbers $\alpha$ and $\beta$, define $$\vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z},\vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x},\text{and}\vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y}.$$ If the vectors $\vec{X},\vec{Y}$, and $\vec{Z}$ lie in a plane, then the value of $\alpha +\beta -3$ is ____.

Let $\gamma \in \mathbb{R}$ be such that the lines $L_1:\frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}$ and $L_2:\frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma }$ intersect. Let $R_1$ be the point of intersection of $L_1$ and $L_2$. Let $O=(0,0,0)$, and $\hat{n}$ denote a unit normal vector to the plane containing both the lines $L_1$ and $L_2$. Match each entry in List-I to the correct entry in List-II. List-I: (P) $\gamma$ equals (Q) A possible choice for $\hat{n}$ is (R) $\overline{O{R}_1}$ equals (S) A possible value of $\overline{O{R}_1}\cdot \hat{n}$ is List-II: (1) $-\hat{i}-\hat{j}+\hat{k}$ (2) $\sqrt{\frac{3}{2}}$ (3) 1 (4) $\frac{1}{\sqrt{6}}\hat{i}-\frac{2}{\sqrt{6}}\hat{j}+\frac{1}{\sqrt{6}}\hat{k}$ (5) $\sqrt{\frac{2}{3}}$ The correct option is

1. A.(P) $\to$ (3) \quad (Q) $\to$ (4) \quad (R) $\to$ (1) \quad (S) $\to$ (2)
2. B.(P) $\to$ (5) \quad (Q) $\to$ (4) \quad (R) $\to$ (1) \quad (S) $\to$ (2)
3. C.(P) $\to$ (3) \quad (Q) $\to$ (4) \quad (R) $\to$ (1) \quad (S) $\to$ (5)
4. D.(P) $\to$ (3) \quad (Q) $\to$ (1) \quad (R) $\to$ (4) \quad (S) $\to$ (5)

Let $\vec{p}=2\hat{i}+\hat{j}+3\hat{k}$ and $\vec{q}=\hat{i}-\hat{j}+\hat{k}$. If for some real numbers $\alpha ,\beta$, and $\gamma$, we have $$15\hat{i}+10\hat{j}+6\hat{k}=\alpha (2\vec{p}+\vec{q})+\beta (\vec{p}-2\vec{q})+\gamma (\vec{p}\times \vec{q}),$$ then the value of $\gamma$ is ____.

Let the position vectors of the points $P,Q,R$ and $S$ be $\vec{a}=\hat{i}+2\hat{j}-5\hat{k}$, $\vec{b}=3\hat{i}+6\hat{j}+3\hat{k}$, $\vec{c}=\frac{17}{5}\hat{i}+\frac{16}{5}\hat{j}+7\hat{k}$ and $\vec{d}=2\hat{i}+\hat{j}+\hat{k}$, respectively. Then which of the following statements is true?

1. A.The points $P,Q,R$ and $S$ are NOT coplanar
2. B.$\frac{\vec{b}+2\vec{d}}{3}$ is the position vector of a point which divides $PR$ internally in the ratio $5:4$
3. C.$\frac{\vec{b}+2\vec{d}}{3}$ is the position vector of a point which divides $PR$ externally in the ratio $5:4$
4. D.The square of the magnitude of the vector $\vec{b}\times \vec{d}$ is 95