Show that the points A, B, C with position vector \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k } \) and \(3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.

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points on the plane

ABCD is a quadrilateral with \(\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha } \) and \(\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta } \) and \(\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta } \). If the area of the quadrilateral is λ times the area of the parallelogram with \(\overset { \rightarrow }{ AB } \) and \(\overset { \rightarrow }{ AD } \) as adjacent sides, then prove that \(\lambda =\frac { 5 }{ 2 } \)

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plane

If \(\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k } \) are two given vector, then find a vector B satisfying the equations \(\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B } \)= \(\overset { \rightarrow }{ C } \) and \(\overset { \rightarrow }{ A } \).\(\overset { \rightarrow }{ B } \) = 3

Find the shortest distance between the following pairs of lines \(\frac { x-3 }{ 3 } =\frac { y-8 }{ -1 } =\frac { z-3 }{ 1 } \)and \(\frac { x+3 }{ -3 } =\frac { y+7 }{ 2 } =\frac { z-6 }{ 4 } \) 

Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

If \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \hat{\mathrm{b}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{i}} \text { and } \overrightarrow{\mathrm{c}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}\) verify that \(\vec{a} \times(\vec{b} \times \vec{c})=(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}\)

Find the shortest distance between the straight lines \(\frac{x-6}{1}=\frac{2-y}{2}=\frac{z-2}{2} \text { and } \frac{x-4}{3}=\frac{y}{-2}=\frac{1-z}{2}\)

Find the non-parametric and Cartesian equations of the plane passing through the point (4, 2, 4) and is perpendicular to the planes 2x + 5y + 4z + 6 = 0 and 4x +7y +62 + 2 = 0.

Find the vector and Cartesian equations of the plane passing through rhe points (2, 2, -11), (3, 4, 2) and (7, 0, 6).

Find the vector and Cartesian equations of the plane through the point (2, -1, -3) and parallel to the lines \(\frac{x-2}{3}=\frac{y-1}{2}=\frac{z-3}{-4} \text { and } \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z-2}{2} \text {. }\)

The foot of perpendicular drawn from the origin to the plane is (4, -2, -5), find the equation of the Plane.

Find the vector and cartesian equatio.n of a plane which is at a distance of 8 units from the orgin and which is normal to the vector \(3 \hat{i}+2 \hat{j}-2 \hat{k}\)

Verify that \((\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})+(\vec{b} \times \vec{c}) \cdot(\vec{a} \times \vec{d})+(\vec{c} \times \vec{a}) \cdot(\vec{b} \times \vec{d})=0\)

Verify \((\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})=[\vec{a}, \vec{b}, \vec{d}] \vec{c}-[\vec{a}, \vec{b}, \vec{c}] \vec{d}\) for \( \vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=2 \hat{i}+\hat{k}, \vec{c}=2 \hat{i}+\hat{j}+\hat{k}, \) \( \vec{d}=\hat{i}+\hat{j}+2 \hat{k}\)

Find \((\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})\) if \( \vec{a} m=\hat{i}+\hat{j}+\hat{k}\), \(\vec{b}=2 \hat{i}+\hat{k}, \ \vec{c}=2 \hat{i}+\hat{j}+\hat{k}, \ \vec{d}=\hat{i}+\hat{j}+2 \hat{k}\)

If \(\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}, \vec{b}=-2 \hat{i}+5 \hat{k}, \vec{c}=\hat{j}-3 \hat{k}\). Verify that  \(\vec{a} \times(\vec{b} \times \vec{c})=(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}\)

For any three vectors \(\vec{a}, \vec{b}, \vec{c}\) prove that \([\vec{a}+\vec{b}, \vec{b}+\vec{c}, \vec{c}+\vec{a}]=2[\vec{a}, \vec{b}, \vec{c}]\)

If the position vectors of three points A, B and, C arc respectively \(\hat{i}+2 \hat{j}+3 \hat{k}, 4 \hat{i}+\hat{i}+5 \hat{k}\) and \(7(\hat{i}+\hat{k})\). Find \(\overrightarrow{A B} \times \overrightarrow{A C}\). Interprer the result geometrically.

If \(\vec{p}=-3 \hat{i}+4 \hat{j}-7 \hat{k}\)  and \(\vec{q}=6 \hat{i}+2 \hat{j}-3 \hat{k}\) then find \(\vec{p} \times \vec{q}\). Verify that \(\vec p\) and \(\vec p \times \vec q\) are perpendicular to each other and also verify that \(\vec q\) and \(\vec{p} \times \vec{q}\) are perpendicular to each other,

Find the vectors of magnitude 6 which are perpendicular to both the vectors \(4 \hat{i}-\hat{j}+3 \hat{k}\) and \(-2 \hat{i}+\hat{j}-2 \hat{k}\)

Find the work done in moving a particle from the point A with position vector \(2 \hat{i}-6 \hat{j}+7 \hat{k}\) to the point B, with position vector \(3 \hat{i}-\hat{j}-5 \hat{k}\), by a force \(\vec{F}=\hat{i}+3 \hat{j}-\hat{k}\)

The work done by the force \(\vec{F}=a \hat{i}+\hat{j}+\hat{k}\) in moving the point of application from (1, 1, 1) to (2, 2, 2) along a straight line is given to be 5 units. Find the value of a,

Find the angle between the vectors \(2 \hat{i}+\hat{j}-\hat{k}\) and \(\hat{i}+2 \hat{j}+\hat{k}\) by using cross product.

A force given by \(3 \hat{i}+2 \hat{j}-4 \hat{k}\) is applied at the point (1, -1, 2). Find the momment of the force about the point (2, -1, 3).