87822 \(O A B C D\) is a pentagon in which the sides \(O A\) and \(C B\) are parallel and the sides \(O D\) and \(A B\) are parallel. Also, it is given that \(\frac{O A}{C B}=2, \frac{O D}{A B}=\frac{1}{3}\). If \(O A=a, O D=d\), then \(\mathbf{A D}+\mathbf{O C}+\mathbf{D C}=\)

87823 If the points
\(\overrightarrow{\mathbf{P}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}, \overrightarrow{\mathrm{Q}}=4 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}, \overrightarrow{\mathrm{R}}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}, \overrightarrow{\mathrm{S}}=\mathbf{a} \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}\)
are the consecutive vertices of a parallelogram PQRS, then

87824 If \(\vec{a}=4 \hat{i}+6 \hat{j}, \vec{b}=3 \hat{j}+4 \hat{k}\) and \(c\) is the projection vector of \(a\) on \(b\), then \(c\) and \(|c|\) respectively are

87825 If \(\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=2 \hat{i}+3 \hat{j}+\hat{k}\),
\(\vec{c}=8 \hat{i}+13 \hat{j}+9 \hat{k}\) and \(x \vec{a}+y \vec{b}+z \vec{c}=0\)
Then \(\frac{\mathrm{xy}}{\mathrm{z}^2}=\)

87822 \(O A B C D\) is a pentagon in which the sides \(O A\) and \(C B\) are parallel and the sides \(O D\) and \(A B\) are parallel. Also, it is given that \(\frac{O A}{C B}=2, \frac{O D}{A B}=\frac{1}{3}\). If \(O A=a, O D=d\), then \(\mathbf{A D}+\mathbf{O C}+\mathbf{D C}=\)

87823 If the points
\(\overrightarrow{\mathbf{P}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}, \overrightarrow{\mathrm{Q}}=4 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}, \overrightarrow{\mathrm{R}}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}, \overrightarrow{\mathrm{S}}=\mathbf{a} \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}\)
are the consecutive vertices of a parallelogram PQRS, then

87824 If \(\vec{a}=4 \hat{i}+6 \hat{j}, \vec{b}=3 \hat{j}+4 \hat{k}\) and \(c\) is the projection vector of \(a\) on \(b\), then \(c\) and \(|c|\) respectively are

87825 If \(\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=2 \hat{i}+3 \hat{j}+\hat{k}\),
\(\vec{c}=8 \hat{i}+13 \hat{j}+9 \hat{k}\) and \(x \vec{a}+y \vec{b}+z \vec{c}=0\)
Then \(\frac{\mathrm{xy}}{\mathrm{z}^2}=\)

87822 \(O A B C D\) is a pentagon in which the sides \(O A\) and \(C B\) are parallel and the sides \(O D\) and \(A B\) are parallel. Also, it is given that \(\frac{O A}{C B}=2, \frac{O D}{A B}=\frac{1}{3}\). If \(O A=a, O D=d\), then \(\mathbf{A D}+\mathbf{O C}+\mathbf{D C}=\)

87823 If the points
\(\overrightarrow{\mathbf{P}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}, \overrightarrow{\mathrm{Q}}=4 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}, \overrightarrow{\mathrm{R}}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}, \overrightarrow{\mathrm{S}}=\mathbf{a} \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}\)
are the consecutive vertices of a parallelogram PQRS, then

87824 If \(\vec{a}=4 \hat{i}+6 \hat{j}, \vec{b}=3 \hat{j}+4 \hat{k}\) and \(c\) is the projection vector of \(a\) on \(b\), then \(c\) and \(|c|\) respectively are

87825 If \(\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=2 \hat{i}+3 \hat{j}+\hat{k}\),
\(\vec{c}=8 \hat{i}+13 \hat{j}+9 \hat{k}\) and \(x \vec{a}+y \vec{b}+z \vec{c}=0\)
Then \(\frac{\mathrm{xy}}{\mathrm{z}^2}=\)

87822 \(O A B C D\) is a pentagon in which the sides \(O A\) and \(C B\) are parallel and the sides \(O D\) and \(A B\) are parallel. Also, it is given that \(\frac{O A}{C B}=2, \frac{O D}{A B}=\frac{1}{3}\). If \(O A=a, O D=d\), then \(\mathbf{A D}+\mathbf{O C}+\mathbf{D C}=\)

87823 If the points
\(\overrightarrow{\mathbf{P}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}, \overrightarrow{\mathrm{Q}}=4 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}, \overrightarrow{\mathrm{R}}=5 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}, \overrightarrow{\mathrm{S}}=\mathbf{a} \hat{\mathbf{i}}+\mathbf{b} \hat{\mathbf{j}}\)
are the consecutive vertices of a parallelogram PQRS, then

87824 If \(\vec{a}=4 \hat{i}+6 \hat{j}, \vec{b}=3 \hat{j}+4 \hat{k}\) and \(c\) is the projection vector of \(a\) on \(b\), then \(c\) and \(|c|\) respectively are

87825 If \(\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=2 \hat{i}+3 \hat{j}+\hat{k}\),
\(\vec{c}=8 \hat{i}+13 \hat{j}+9 \hat{k}\) and \(x \vec{a}+y \vec{b}+z \vec{c}=0\)
Then \(\frac{\mathrm{xy}}{\mathrm{z}^2}=\)