A tetrahedron has vertices at $O\left( {0,0,0} \right),$ $A\left( {1,2,1} \right),\;B\left( {2,1,3} \right){\rm{and\;}}C\left( { - 1,1,2} \right).$ Then, the angle between the faces $OAB$ and $ABC$ will be

${\rm{co}}{{\rm{s}}^{ - 1}}\left( {\frac{{19}}{{35}}} \right)$

${\rm{co}}{{\rm{s}}^{ - 1}}\left( {\frac{{17}}{{31}}} \right)$

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JEE • MATHEMATICS • CROSS PRODUCT

Cross product practice · Hard

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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1 Hard

1 of 5

Application +4 | -1

A rigid body rotates with constant angular velocity

\omega

about the line whose vector equation is,

\mathbf{r}=\lambda(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})

. The speed of the particle at the instant it passes through the point with position vector

(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})

is equal to

View answer and explanation

Correct option: A

\omega \sqrt{2}

Option A (

\omega \sqrt{2}

) is correct because it represents the speed of the particle, calculated using the cross product of angular velocity and position vector.

2 Hard

2 of 5

Application +4 | -1

Let

\vec{a}=\alpha_1 \hat{i}+\alpha_2 \hat{j}+\alpha_3 \hat{k}, \vec{b}=\beta_1 \hat{i}+\beta_2 \hat{j}+\beta_3 \hat{k}

and

\vec{c}=\gamma_1 \hat{i}+\gamma_2 \hat{j}+\gamma_3 \hat{k}

and

|\vec{a}|=2 \sqrt{2}

makes the angle

\pi / 3

with the plane of

\overrightarrow{\mathrm{b}}

and

\overrightarrow{\mathrm{c}}

and angle between

\overrightarrow{\mathrm{b}} \& \overrightarrow{\mathrm{c}}

\pi / 6

then the value of

\left|\begin{array}{lll}\alpha_1 & \alpha_2 & \alpha_3 \\ \beta_1 & \beta_2 & \beta_3 \\ \gamma_1 & \gamma_2 & \gamma_3\end{array}\right|

equals

View answer and explanation

Correct option: C

\left(\frac{\sqrt{3}(|\overrightarrow{\mathrm{b}}| \overrightarrow{\mathrm{c}} \mid)}{\sqrt{2}}\right)^{\mathrm{n}}

Option C (

\left(\frac{{\sqrt{{3}}(|\overrightarrow{{\mathrm{{b}}}}||\overrightarrow{{\mathrm{{c}}}}|)}}{{\sqrt{{2}}}}\right)^{\mathrm{{n}}}

) is correct because it correctly represents the magnitude of the scalar triple product of the vectors in terms of their magnitudes and the angles between them.

3 Hard

3 of 5

Analyzing +4 | -1

{\vec a}

and

{\vec b}

are not perpendicular to each other and

\vec r \times \vec a = \vec b \times \vec a,\;\vec r.\vec c = 0

, then

{\vec r}

is equal to

View answer and explanation

Correct option: C

\vec b - \frac{{\left( {\vec b.\vec c} \right)}}{{\left( {\vec a.\vec c} \right)}}\vec a

Option C (

\vec b - \frac{{\left( {\vec b.\vec c} \right)}}{{\left( {\vec a.\vec c} \right)}}\vec a

) is correct because it satisfies both the cross product and dot product conditions.

4 Hard

4 of 5

Application +4 | -1

A tetrahedron has vertices at

O\left( {0,0,0} \right),

A\left( {1,2,1} \right),\;B\left( {2,1,3} \right)\mathrm{and\;}C\left( { - 1,1,2} \right).

Then, the angle between the faces

OAB

and

ABC

will be

View answer and explanation

Correct option: A

\mathrm{co}{\mathrm{s}^{ - 1}}\left( {\frac{{19}}{{35}}} \right)

Option A (

\mathrm{co}{\mathrm{s}^{ - 1}}(\frac{{19}}{{35}}))

is correct because the angle between two faces is found by calculating the dot product of their normal vectors.

5 Hard

5 of 5

Application +4 | -1

If vector

\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}

is rotated through an angle of

90^{\circ}

, so as to cross the positive direction of

Y

-axis, then the vector in the new position is

View answer and explanation

Correct option: A

-\frac{2}{\sqrt{5}} \hat{\mathbf{i}}+\sqrt{5} \hat{\mathbf{j}}-\frac{4}{\sqrt{5}} \hat{\mathbf{k}}

Option A (

\frac{-2}{\sqrt{5}} \hat{\mathbf{i}}+\sqrt{5} \hat{\mathbf{j}}-\frac{4}{\sqrt{5}} \hat{\mathbf{k}}

) is correct because it represents the vector after rotation, satisfying the given conditions of crossing the Y-axis and rotating by 90 degrees.

Revision notes and key takeaways

Review CROSS PRODUCT in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 5 single-correct-answer MCQs on CROSS PRODUCT.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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Cross product practice · Hard

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