121196 If the points \((1,1, \lambda)\) and \((-3,0,1)\) are equidistant from the plane \(\overrightarrow{\mathbf{r}} \cdot(\mathbf{3 i}+\mathbf{4} \mathbf{j}-12 \hat{k})=-13\) then \(\lambda=\)

121202 Let \(\vec{\alpha}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\boldsymbol{\beta}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\vec{\gamma}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) be three vectors. A vector \(\vec{\delta}\) in the plane of \(\vec{\alpha}\) and \(\vec{\beta}\), whose projection on \(\vec{\gamma}\) is \(\frac{1}{\sqrt{3}}\), is given by

121208 Let the plane \(P: \vec{r} \cdot \vec{a}=d\) contain the line of intersection of two planes \(\overrightarrow{\mathbf{r}} \cdot(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}})=6\) and \(\overrightarrow{\mathbf{r}} \cdot(-6 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-\hat{\mathbf{k}})=7\). If the plane \(P\) passes through the point \(\left(2,3, \frac{1}{2}\right)\), then the value of \(\frac{|13 \vec{a}|^2}{d^2}\) is equal to

121187 Let \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}},\)
\(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\), a vector in the plane of \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\) whose projection on \(\overrightarrow{\mathbf{c}}\) is \(\frac{1}{\sqrt{3}}\) is

121196 If the points \((1,1, \lambda)\) and \((-3,0,1)\) are equidistant from the plane \(\overrightarrow{\mathbf{r}} \cdot(\mathbf{3 i}+\mathbf{4} \mathbf{j}-12 \hat{k})=-13\) then \(\lambda=\)

121202 Let \(\vec{\alpha}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\boldsymbol{\beta}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\vec{\gamma}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) be three vectors. A vector \(\vec{\delta}\) in the plane of \(\vec{\alpha}\) and \(\vec{\beta}\), whose projection on \(\vec{\gamma}\) is \(\frac{1}{\sqrt{3}}\), is given by

121208 Let the plane \(P: \vec{r} \cdot \vec{a}=d\) contain the line of intersection of two planes \(\overrightarrow{\mathbf{r}} \cdot(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}})=6\) and \(\overrightarrow{\mathbf{r}} \cdot(-6 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-\hat{\mathbf{k}})=7\). If the plane \(P\) passes through the point \(\left(2,3, \frac{1}{2}\right)\), then the value of \(\frac{|13 \vec{a}|^2}{d^2}\) is equal to

121187 Let \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}},\)
\(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\), a vector in the plane of \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\) whose projection on \(\overrightarrow{\mathbf{c}}\) is \(\frac{1}{\sqrt{3}}\) is

121196 If the points \((1,1, \lambda)\) and \((-3,0,1)\) are equidistant from the plane \(\overrightarrow{\mathbf{r}} \cdot(\mathbf{3 i}+\mathbf{4} \mathbf{j}-12 \hat{k})=-13\) then \(\lambda=\)

121202 Let \(\vec{\alpha}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\boldsymbol{\beta}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\vec{\gamma}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) be three vectors. A vector \(\vec{\delta}\) in the plane of \(\vec{\alpha}\) and \(\vec{\beta}\), whose projection on \(\vec{\gamma}\) is \(\frac{1}{\sqrt{3}}\), is given by

121208 Let the plane \(P: \vec{r} \cdot \vec{a}=d\) contain the line of intersection of two planes \(\overrightarrow{\mathbf{r}} \cdot(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}})=6\) and \(\overrightarrow{\mathbf{r}} \cdot(-6 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-\hat{\mathbf{k}})=7\). If the plane \(P\) passes through the point \(\left(2,3, \frac{1}{2}\right)\), then the value of \(\frac{|13 \vec{a}|^2}{d^2}\) is equal to

121187 Let \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}},\)
\(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\), a vector in the plane of \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\) whose projection on \(\overrightarrow{\mathbf{c}}\) is \(\frac{1}{\sqrt{3}}\) is

121196 If the points \((1,1, \lambda)\) and \((-3,0,1)\) are equidistant from the plane \(\overrightarrow{\mathbf{r}} \cdot(\mathbf{3 i}+\mathbf{4} \mathbf{j}-12 \hat{k})=-13\) then \(\lambda=\)

121202 Let \(\vec{\alpha}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\boldsymbol{\beta}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\vec{\gamma}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) be three vectors. A vector \(\vec{\delta}\) in the plane of \(\vec{\alpha}\) and \(\vec{\beta}\), whose projection on \(\vec{\gamma}\) is \(\frac{1}{\sqrt{3}}\), is given by

121208 Let the plane \(P: \vec{r} \cdot \vec{a}=d\) contain the line of intersection of two planes \(\overrightarrow{\mathbf{r}} \cdot(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}})=6\) and \(\overrightarrow{\mathbf{r}} \cdot(-6 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-\hat{\mathbf{k}})=7\). If the plane \(P\) passes through the point \(\left(2,3, \frac{1}{2}\right)\), then the value of \(\frac{|13 \vec{a}|^2}{d^2}\) is equal to

121187 Let \(\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}},\)
\(\overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\), a vector in the plane of \(\overrightarrow{\mathbf{a}}\) and \(\overrightarrow{\mathbf{b}}\) whose projection on \(\overrightarrow{\mathbf{c}}\) is \(\frac{1}{\sqrt{3}}\) is