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$2\hat{a}$
Explore More
Similar Questions
Column-$I$ Column-$II$ $(A)$ In $R^2$,if the magnitude of the projection vector of the vector $\alpha \hat{i}+\beta \hat{j}$ on $\sqrt{3} \hat{i}+\hat{j}$ is $\sqrt{3}$ and if $\alpha=2+\sqrt{3} \beta$,then possible value$(s)$ of $|\alpha|$ is (are) $(P)$ $1$ $(B)$ Let $a$ and $b$ be real numbers such that the function $f(x)=\begin{cases} -3ax^2-2, & x < 1 \\ bx+a^2, & x \geq 1 \end{cases}$ is differentiable for all $x \in R$. Then possible value$(s)$ of $a$ is (are) $(Q)$ $2$ $(C)$ Let $\omega \neq 1$ be a complex cube root of unity. If $(3-3\omega+2\omega^2)^{4n+3} + (2+3\omega-3\omega^2)^{4n+3} + (-3+2\omega+3\omega^2)^{4n+3}=0$,then possible value$(s)$ of $n$ is (are) $(R)$ $3$ $(D)$ Let the harmonic mean of two positive real numbers $a$ and $b$ be $4$. If $q$ is a positive real number such that $a, 5, q, b$ is an arithmetic progression,then the value$(s)$ of $|q-a|$ is (are) $(S)$ $4$ $(T)$ $5$
| Column-$I$ | Column-$II$ |
| $(A)$ In $R^2$,if the magnitude of the projection vector of the vector $\alpha \hat{i}+\beta \hat{j}$ on $\sqrt{3} \hat{i}+\hat{j}$ is $\sqrt{3}$ and if $\alpha=2+\sqrt{3} \beta$,then possible value$(s)$ of $|\alpha|$ is (are) | $(P)$ $1$ |
| $(B)$ Let $a$ and $b$ be real numbers such that the function $f(x)=\begin{cases} -3ax^2-2, & x < 1 \\ bx+a^2, & x \geq 1 \end{cases}$ is differentiable for all $x \in R$. Then possible value$(s)$ of $a$ is (are) | $(Q)$ $2$ |
| $(C)$ Let $\omega \neq 1$ be a complex cube root of unity. If $(3-3\omega+2\omega^2)^{4n+3} + (2+3\omega-3\omega^2)^{4n+3} + (-3+2\omega+3\omega^2)^{4n+3}=0$,then possible value$(s)$ of $n$ is (are) | $(R)$ $3$ |
| $(D)$ Let the harmonic mean of two positive real numbers $a$ and $b$ be $4$. If $q$ is a positive real number such that $a, 5, q, b$ is an arithmetic progression,then the value$(s)$ of $|q-a|$ is (are) | $(S)$ $4$ |
| $(T)$ $5$ |
DifficultIIT 2015
View SolutionLet $\vec{a}$ and $\vec{b}$ be two unit vectors such that $|\vec{a} + \vec{b}| = \sqrt{3}$. If $\vec{c} = \vec{a} + 2\vec{b} + 3(\vec{a} \times \vec{b})$,then $2|\vec{c}|$ is equal to
DifficultJEE MAIN 2015
View SolutionIf $b$ and $c$ are non-collinear vectors,$|c| \neq 0$,$a \times(b \times c)+(a \cdot b) b=(4-2 \beta-\sin \alpha) b+\left(\beta^2-1\right) c$ and $(c \cdot c) a=c$,then the scalars $\alpha$ and $\beta$ are
Observe the following lists. Then the correct match for List-$I$ from List-$II$ is:
List-$I$ List-$II$ $(A)$ $[\mathbf{a} \mathbf{b} \mathbf{c}]$ $1. |\mathbf{a}||\mathbf{b}|\cos(\mathbf{a}, \mathbf{b})$ $(B)$ $(\mathbf{c} \times \mathbf{a}) \times \mathbf{b}$ $2. (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$ $(C)$ $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$ $3. \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$ $(D)$ $\mathbf{a} \cdot \mathbf{b}$ $4. |\mathbf{a}||\mathbf{b}|$ $5. (\mathbf{b} \cdot \mathbf{c})\mathbf{a} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$
| List-$I$ | List-$II$ |
| $(A)$ $[\mathbf{a} \mathbf{b} \mathbf{c}]$ | $1. |\mathbf{a}||\mathbf{b}|\cos(\mathbf{a}, \mathbf{b})$ |
| $(B)$ $(\mathbf{c} \times \mathbf{a}) \times \mathbf{b}$ | $2. (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$ |
| $(C)$ $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$ | $3. \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$ |
| $(D)$ $\mathbf{a} \cdot \mathbf{b}$ | $4. |\mathbf{a}||\mathbf{b}|$ |
| $5. (\mathbf{b} \cdot \mathbf{c})\mathbf{a} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$ |
DifficultTS EAMCET 2005
View SolutionIf $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}, \overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}$ 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) $\overrightarrow{b} \times \overrightarrow{c}$ | $(D)$ $2\hat{i}-2\hat{k}$ |
| $(E)$ $2\hat{j}+2\hat{k}$ | |
| $(F)$ $4$ |
Vedclass Products
For Students
Vedclass Test Series
Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.
Start Free TrialFor Teachers
Exam Paper Generator
Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.
Try FreeFor Institutes
Online Exam Module
Live online exams with unlimited students, 360° analytics & white-label branding.
See Demo