let alpha beta gamma be distinct real numbers the points

Let $$\alpha ,\,\beta ,\,\gamma $$ be distinct real numbers. The points with position vectors $$\alpha \hat i + \beta \hat j + \gamma \hat k,\,\beta \hat i + \gamma \hat j + \alpha \hat k$$ and $$\gamma \hat i + \alpha \hat j + \beta \hat k$$

A. are collinear

B. form an equilateral triangle

C. form a scalene triangle

D. form a right-angled triangle

Answer : are collinear

Solution :
$$\alpha ,\,\beta $$ and $$\gamma $$ be distinct real numbers
$$\alpha \hat i + \beta \hat j + \gamma \hat k\,;\,\beta \hat i + \gamma \hat j + \alpha \hat k\,;\,\gamma \hat i + \alpha \hat j + \beta \hat k$$
$$\overrightarrow a ,\,\overrightarrow b $$ and $$\overrightarrow c $$ are collinear
If $$a = \alpha ,\,b = \beta ,\,c = \gamma \,\,\,\left( {\because \,\alpha = \hat i + \hat j + \hat k} \right)$$

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 $$\alpha ,\,\beta ,\,\gamma $$ be distinct real numbers. The points with position vectors $$\alpha \hat i + \beta \hat j + \gamma \hat k,\,\beta \hat i + \gamma \hat j + \alpha \hat k$$ and $$\gamma \hat i + \alpha \hat j + \beta \hat k$$

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

Practice More MCQ Question on Maths Section

Let $$\alpha ,\,\beta ,\,\gamma $$ be distinct real numbers. The points with position vectors $$\alpha \hat i + \beta \hat j + \gamma \hat k,\,\beta \hat i + \gamma \hat j + \alpha \hat k$$ and $$\gamma \hat i + \alpha \hat j + \beta \hat k$$

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 :

Practice More Releted MCQ Question on 3D Geometry and Vectors

Practice More MCQ Question on Maths Section

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

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3D Geometry and Vectors