If $\hat{a}, \hat{b}$ and $\hat{c}$ are non-coplanar vectors and if $\hat{d}$ is such that $\hat{d} = \frac{1}{x}(\hat{a} + \hat{b} + \hat{c})$ and $\hat{d} = \frac{1}{y}(\hat{b} + \hat{c} + \hat{d})$ where $x$ and $y$ are non-zero real numbers,then $\frac{1}{xy}(\hat{a} + \hat{b} + \hat{c} + \hat{d})$ equals to
- A$3\hat{c}$
- B$-\hat{a}$
- C$0$
- D$2\hat{a}$
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$A, B, P, Q, R$ are five points in a plane. If the forces acting at point $A$ are $\overline{AP}, \overline{AQ}, \overline{AR}$ and the forces acting at point $B$ are $\overline{PB}, \overline{QB}, \overline{RB}$,find the resultant of all these forces.
Medium
View SolutionThe vector that is parallel to the vector $2 \hat{i} - 2 \hat{j} - 4 \hat{k}$ and coplanar with the vectors $\hat{i} + \hat{j}$ and $\hat{j} + \hat{k}$ is
If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}, \overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{d}=\hat{i}-\hat{j}-\hat{k}$, then match the following List-$I$ with List-$II$:
List-$I$ List-$II$ $(i)$ $\overrightarrow{a} \cdot \overrightarrow{b}$ $(A)$ $\overrightarrow{a} \cdot \overrightarrow{d}$ (ii) $\overrightarrow{b} \cdot \overrightarrow{c}$ $(B)$ $3$ (iii) $[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$ $(C)$ $\overrightarrow{b} \cdot \overrightarrow{d}$ (iv) $\overrightarrow{b} \times \overrightarrow{c}$ $(D)$ $2\hat{i}-2\hat{k}$ $(E)$ $2\hat{j}+2\hat{k}$ $(F)$ $4$
| List-$I$ | List-$II$ |
| $(i)$ $\overrightarrow{a} \cdot \overrightarrow{b}$ | $(A)$ $\overrightarrow{a} \cdot \overrightarrow{d}$ |
| (ii) $\overrightarrow{b} \cdot \overrightarrow{c}$ | $(B)$ $3$ |
| (iii) $[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$ | $(C)$ $\overrightarrow{b} \cdot \overrightarrow{d}$ |
| (iv) $\overrightarrow{b} \times \overrightarrow{c}$ | $(D)$ $2\hat{i}-2\hat{k}$ |
| $(E)$ $2\hat{j}+2\hat{k}$ | |
| $(F)$ $4$ |
Match List $I$ with List $II$ and select the correct answer using the code given below the lists:
List $I$ List $II$ $P$. Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $2$. Then the volume of the parallelepiped determined by vectors $2(\vec{a} \times \vec{b}), 3(\vec{b} \times \vec{c})$ and $(\vec{c} \times \vec{a})$ is $1$. $100$ $Q$. Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $5$. Then the volume of the parallelepiped determined by vectors $3(\vec{a}+\vec{b}), (\vec{b}+\vec{c})$ and $2(\vec{c}+\vec{a})$ is $2$. $30$ $R$. Area of a triangle with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $20$. Then the area of the triangle with adjacent sides determined by vectors $(2\vec{a}+3\vec{b})$ and $(\vec{a}-\vec{b})$ is $3$. $24$ $S$. Area of a parallelogram with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $30$. Then the area of the parallelogram with adjacent sides determined by vectors $(\vec{a}+\vec{b})$ and $\vec{a}$ is $4$. $60$
Codes: $P \quad Q \quad R \quad S$
| List $I$ | List $II$ |
| $P$. Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $2$. Then the volume of the parallelepiped determined by vectors $2(\vec{a} \times \vec{b}), 3(\vec{b} \times \vec{c})$ and $(\vec{c} \times \vec{a})$ is | $1$. $100$ |
| $Q$. Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $5$. Then the volume of the parallelepiped determined by vectors $3(\vec{a}+\vec{b}), (\vec{b}+\vec{c})$ and $2(\vec{c}+\vec{a})$ is | $2$. $30$ |
| $R$. Area of a triangle with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $20$. Then the area of the triangle with adjacent sides determined by vectors $(2\vec{a}+3\vec{b})$ and $(\vec{a}-\vec{b})$ is | $3$. $24$ |
| $S$. Area of a parallelogram with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $30$. Then the area of the parallelogram with adjacent sides determined by vectors $(\vec{a}+\vec{b})$ and $\vec{a}$ is | $4$. $60$ |
Codes: $P \quad Q \quad R \quad S$
DifficultIIT 2013
View SolutionMatch the statements given in Column $I$ with the values given in Column $II$.
Column $I$ Column $II$ $(A)$ If $\vec{a}=\hat{j}+\sqrt{3} \hat{k}, \vec{b}=-\hat{j}+\sqrt{3} \hat{k}$ and $\vec{c}=2 \sqrt{3} \hat{k}$ form a triangle,then the internal angle of the triangle between $\vec{a}$ and $\vec{b}$ is $(p)$ $\frac{\pi}{6}$ $(B)$ If $\int_a^b(f(x)-3 x) d x=a^2-b^2$,then the value of $f\left(\frac{\pi}{6}\right)$ is $(q)$ $\frac{2 \pi}{3}$ $(C)$ The value of $\frac{\pi^2}{\ln 3} \int_{1 / 6}^{5 / 6} \sec (\pi x) d x$ is $(r)$ $\frac{\pi}{3}$ $(D)$ The maximum value of $|\operatorname{Arg}(\frac{1}{1-z})|$ for $|z|=1, z \neq 1$ is given by $(s)$ $\pi$ $(t)$ $\frac{\pi}{2}$
| Column $I$ | Column $II$ |
| $(A)$ If $\vec{a}=\hat{j}+\sqrt{3} \hat{k}, \vec{b}=-\hat{j}+\sqrt{3} \hat{k}$ and $\vec{c}=2 \sqrt{3} \hat{k}$ form a triangle,then the internal angle of the triangle between $\vec{a}$ and $\vec{b}$ is | $(p)$ $\frac{\pi}{6}$ |
| $(B)$ If $\int_a^b(f(x)-3 x) d x=a^2-b^2$,then the value of $f\left(\frac{\pi}{6}\right)$ is | $(q)$ $\frac{2 \pi}{3}$ |
| $(C)$ The value of $\frac{\pi^2}{\ln 3} \int_{1 / 6}^{5 / 6} \sec (\pi x) d x$ is | $(r)$ $\frac{\pi}{3}$ |
| $(D)$ The maximum value of $|\operatorname{Arg}(\frac{1}{1-z})|$ for $|z|=1, z \neq 1$ is given by | $(s)$ $\pi$ |
| $(t)$ $\frac{\pi}{2}$ |
DifficultIIT 2011
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