let pqr and s be the points on the plane with position

Let $$P,\,Q,\,R$$ and $$S$$ be the points on the plane with position vectors $$ - 2\hat i - \hat j,\,4\hat i,\,3\hat i + 3\hat j$$ and $$ - 3\hat i + 2\hat j$$ respectively, The quadrilateral $$PQRS$$ must be a :

A. parallelogram, which is neither a rhombus nor a rectangle

B. square

C. rectangle, but not a square

D. rhombus, but not a square

Answer : parallelogram, which is neither a rhombus nor a rectangle

Solution :
We have $$\overrightarrow {PQ} = 6\hat i + \hat j,\,\overrightarrow {QR} = - \hat i + 3\hat j,\,\overrightarrow {SR} = 6\hat i + \hat j,\,\overrightarrow {PS} = - \hat i + 3\hat j$$
$$\eqalign{ & \Rightarrow \overrightarrow {PQ} = \overrightarrow {SR} \,;\,\overrightarrow {QR} = \overrightarrow {PS} \,\,{\text{and }}\overrightarrow {PQ} .\overrightarrow {PS} \ne 0 \cr & {\text{Also }}\left| {\overrightarrow {PQ} } \right| \ne \left| {\overrightarrow {QR} } \right| \cr} $$
$$ \Rightarrow PQRS$$ is a parallelogram but neither a rhombus nor a rectangle.

Releted MCQ Question on
Geometry >> 3D Geometry and Vectors

Releted Question 1

The scalar $$\vec A.\left( {\vec B + \vec C} \right) \times \left( {\vec A + \vec B + \vec C} \right)$$ equals :

A. $$0$$

B. $$\left[ {\vec A\,\vec B\,\vec C} \right] + \left[ {\vec B\,\vec C\,\vec A} \right]$$

C. $$\left[ {\vec A\,\vec B\,\vec C} \right]$$

D. None of these

View Answer

Releted Question 2

For non-zero vectors $$\vec a,\,\vec b,\,\vec c,\,\left| {\left( {\vec a \times \vec b} \right).\vec c} \right| = \left| {\vec a} \right|\left| {\vec b} \right|\left| {\vec c} \right|$$ holds if and only if -

A. $$\vec a.\vec b = 0,\,\,\,\vec b.\vec c = 0$$

B. $$\vec b.\vec c = 0,\,\,\,\vec c.\vec a = 0$$

C. $$\vec c.\vec a = 0,\,\,\,\vec a.\vec b = 0$$

D. $$\vec a.\vec b = \vec b.\vec c = \vec c.\vec a = 0$$

View Answer

Releted Question 3

The volume of the parallelepiped whose sides are given by $$\overrightarrow {OA} = 2i - 2j,\,\,\overrightarrow {OB} = i + j - k,\,\,\overrightarrow {OC} = 3i - k,$$ is :

A. $$\frac{4}{{13}}$$

B. $$4$$

C. $$\frac{2}{7}$$

D. none of these

View Answer

Releted Question 4

The points with position vectors $$60i + 3j,\,\,40i - 8j,\,\,ai - 52j$$ are collinear if :

A. $$a = - 40$$

B. $$a = 40$$

C. $$a = 20$$

D. none of these

View Answer

Let $$P,\,Q,\,R$$ and $$S$$ be the points on the plane with position vectors $$ - 2\hat i - \hat j,\,4\hat i,\,3\hat i + 3\hat j$$ and $$ - 3\hat i + 2\hat j$$ respectively, The quadrilateral $$PQRS$$ must be a :

Releted MCQ Question on
Geometry >> 3D Geometry and Vectors

The scalar $$\vec A.\left( {\vec B + \vec C} \right) \times \left( {\vec A + \vec B + \vec C} \right)$$ equals :

For non-zero vectors $$\vec a,\,\vec b,\,\vec c,\,\left| {\left( {\vec a \times \vec b} \right).\vec c} \right| = \left| {\vec a} \right|\left| {\vec b} \right|\left| {\vec c} \right|$$ holds if and only if -

The volume of the parallelepiped whose sides are given by $$\overrightarrow {OA} = 2i - 2j,\,\,\overrightarrow {OB} = i + j - k,\,\,\overrightarrow {OC} = 3i - k,$$ is :

The points with position vectors $$60i + 3j,\,\,40i - 8j,\,\,ai - 52j$$ are collinear if :

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3D Geometry and Vectors

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Let $$P,\,Q,\,R$$ and $$S$$ be the points on the plane with position vectors $$ - 2\hat i - \hat j,\,4\hat i,\,3\hat i + 3\hat j$$ and $$ - 3\hat i + 2\hat j$$ respectively, The quadrilateral $$PQRS$$ must be a :

Releted MCQ Question on Geometry >> 3D Geometry and Vectors

The scalar $$\vec A.\left( {\vec B + \vec C} \right) \times \left( {\vec A + \vec B + \vec C} \right)$$ equals :

For non-zero vectors $$\vec a,\,\vec b,\,\vec c,\,\left| {\left( {\vec a \times \vec b} \right).\vec c} \right| = \left| {\vec a} \right|\left| {\vec b} \right|\left| {\vec c} \right|$$ holds if and only if -

The volume of the parallelepiped whose sides are given by $$\overrightarrow {OA} = 2i - 2j,\,\,\overrightarrow {OB} = i + j - k,\,\,\overrightarrow {OC} = 3i - k,$$ is :

The points with position vectors $$60i + 3j,\,\,40i - 8j,\,\,ai - 52j$$ are collinear if :

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Releted MCQ Question on
Geometry >> 3D Geometry and Vectors

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3D Geometry and Vectors