Find the Cartesian equation of a line passing through the points A(2, -1, 3) and B(4, 2, 1)

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-1

Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line \({ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right) \)

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p=-1

Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3 as direction ratios of normal to the plane.

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2

If the planes \({ \overset { \rightarrow }{ r } }.\left( \overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \right) =7\) and \({ \overset { \rightarrow }{ r } }.\left( \lambda \overset { \wedge }{ i } +2\overset { \wedge }{ j } -7\overset { \wedge }{ k } \right) =26\) are perpendicular. Find the value of λ.

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∆=4

Find the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and 2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

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x = -1 is one root

Let \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) be unit vectors such \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ c } =0\) and the angle between \(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) is \(\frac { \pi }{ 6 } \). Prove that \(\overset { \rightarrow }{ a } =\pm 2\left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) \)

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Type I even degree reciprocal equation

Prove that for any two vectors \(\vec{a} \text { and } \vec{b}\)\(|\vec{a}+\vec{b}| \leq|\vec{a}|+|b|\) (Triangle inequality)

Find \(\vec{a} \cdot \vec{b}\) when
\((i)\ \vec{a}=\hat{i}-2 \hat{j}+\hat{k}\ and\ \vec{b}=4 \hat{i}-4 \hat{j}+7 \hat{k} \)
\((ii)\ \vec{a}=\hat{j}+2 \hat{k}\ and\ \vec{b}=2 \hat{i}+\hat{k} \)
\((iii)\ \vec{a}=\hat{j}-2 \hat{k}\ and\ \vec{b}=2 \hat{i}+3 \hat{j}-2 \hat{k} \)

For what value of m the vectors \(\vec a\) and \(\vec b\) perpendicular to each other.
\((i) \ \vec{a}=m \hat{i}+2 \hat{j}+\hat{k}\ and \ \vec{b}=4 \hat{i}-9 \hat{j}+2 \hat{k} \)
\((ii) \ \vec{a}=5 \hat{i}-9 \hat{j}+2 \hat{k}\ and \ \vec{b}=m \hat{i}+2 \hat{j}+\hat{k}\)

lf  \(\vec a\)and \(\vec b\)are two vectors iuch that \(|\vec{a}|=4\), \(|\vec{b}|=3 \text { and } \vec{a} \cdot \vec{b}=6\). Find the angle between \(\vec a \text { and } \vec{b}\)

For any vector \(\vec{r},\) prove that \(\vec{r}=(\vec{r} \cdot \hat{i}) \hat{i}+(\vec{r} \cdot \hat{j}) \hat{j}+(\vec{r} \cdot \hat{k}) \hat{k}\)

Find the projection of the vector \(7 \hat{i}+\hat{j}-4 \hat{k}\) on \(2 \hat{i}+6 \hat{j}+3 \hat{k}\)

If  \(\vec{a} \text { and } \vec{b}\), are unit vectors inclined at an angle \(\theta\), then prove that \(\sin \frac{\theta}{2}=\frac{1}{2}|\vec{a}-\vec{b}|\)

If \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0},|\vec{a}|=3,|\vec{b}|=5 \text { and }|\vec{c}|=7,\) find the angle between \(\vec a \text { and } \vec{b}\)

Prove that \(|\left[\begin{array}{lll} \vec{a} & \vec{b} & \bar{c} \end{array}\right]|=a b c\) if and only if  \(\vec a,\vec{b}, \vec{c}\) are mutually perpendicular.