Vector or Cross Product - Maharashtra Board STD 12 Science Maths Questions

Q 1MCQ1 Mark

If $\theta$ is the angle between the vectors $2\hat{\text{i}}-2\hat{\text{j}}+4\hat{\text{k}}$ and $3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}},$ then $\sin\theta=$

A
$\frac{2}{3}$
✓
$\frac{2}{\sqrt{7}}$
C
$\frac{\sqrt{2}}{7}$
D
$\sqrt{\frac{2}{7}}$

Answer: B.

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Q 2MCQ1 Mark

A unit vector perpendicular to both $\hat{\text{i}}+\hat{\text{j}}$ and $\hat{\text{j}}+\hat{\text{k}}$ is:

A
$\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$
B
$\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$
C
$\frac{1}{\sqrt{3}}\big(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\big)$
✓
$\frac{1}{\sqrt{3}}\big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$

Answer: D.

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Q 3MCQ1 Mark

vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ are inclined at angle $\theta=120^\circ.$ if $|\vec{\text{a}}|=1,\big|\vec{\text{b}}\big|=2,$ then $\big[\big(\vec{\text{a}}+3\vec{\text{b}}\big)\times\big(3\vec{\text{a}}-\vec{\text{b}}\big)\big]^2$ is equal to:

✓
$300$
B
$325$
C
$275$
D
$225$

Answer: A.

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Q 4MCQ1 Mark

If $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=4,\big|\vec{\text{a}}.\vec{\text{b}}\big|=2,$ then $|\vec{\text{a}}|^2\big|\vec{\text{b}}\big|^2=$

A
$6$
B
$2$
✓
$20$
D
$8$

Answer: C.

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Q 5MCQ1 Mark

The value of $\hat{\text{i}}.\big(\hat{\text{j}}\times\hat{\text{k}}\big)+\hat{\text{j}}.\big(\hat{\text{i}}\times\hat{\text{k}}\big)+\hat{\text{k}}.\big(\hat{\text{i}}\times\hat{\text{j}}\big),$ is:

A
$0$
B
$-1$
✓
$1$
D
$3$

Answer: C.

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Q 6Solve the Following Question.(2 Marks)2 Marks

For any two vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ write the value of $\big(\vec{\text{a}}.\vec{\text{b}}\big)^2+\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2$ in terms of their magnitudes.

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Q 7Solve the Following Question.(2 Marks)2 Marks

For any two vectors $\vec{\text{a}}$ and $\vec{\text{b}},$ find $\big(\vec{\text{a}}\times\vec{\text{b}}\big).\vec{\text{b}}.$

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Q 8Solve the Following Question.(2 Marks)2 Marks

If $\vec{\text{a}}=3\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},$ then find $\big(\vec{\text{a}}\times\vec{\text{b}}\big)\vec{\text{a}}.$

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Q 9Solve the Following Question.(2 Marks)2 Marks

If $\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2+\big(\vec{\text{a}}.\vec{\text{b}}\big)^2=144$ and $|\vec{\text{a}}|=4,$ find $\big|\vec{\text{b}}\big|.$

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Q 10Solve the Following Question.(2 Marks)2 Marks

Write the value of $\hat{\text{i}}.\big(\hat{\text{j}}\times\hat{\text{k}}\big)+\hat{\text{j}}.\big(\hat{\text{k}}\times\hat{\text{i}}\big)+\hat{\text{k}}.\big(\hat{\text{i}}\times\hat{\text{j}}\big).$

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Q 11Solve the Following Question.(3 Marks)3 Marks

Find a unit vector perpendicular to both the vectors $4\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $-2\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}.}$

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Q 12Solve the Following Question.(3 Marks)3 Marks

Write the number of vectors of unit length perpendicular to both the vectors $\vec{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}$and $\vec{\text{b}}=\hat{\text{j}}+\hat{\text{k}}.$

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Q 13Solve the Following Question.(3 Marks)3 Marks

If $|\vec{\text{a}}|=\sqrt{26,}\big|\vec{\text{b}}\big|=7$ and $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=35,$ find $\vec{\text{a}}.\vec{\text{b}}.$

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Q 14Solve the Following Question.(3 Marks)3 Marks

Find the area of the parallelogram whose diagonals are:
$4\hat{\text{i}}-\hat{\text{j}}-3\hat{\text{k}}$ and $-2\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}}$

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Q 15Solve the Following Question.(3 Marks)3 Marks

If $\vec{\text{a}}\times\vec{\text{b}}=\vec{\text{b}}\times\vec{\text{c}}\neq0,$ then show that $\vec{\text{a}}+\vec{\text{c}}=\text{m}\vec{\text{b}},$ where m is any scalar.

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Q 16Solve the Following Question.(5 Marks)5 Marks

Find a unit vector perpendicular to the plane A B C, where the coordinates of A, B and C are A (3, -1, 2), B (1, -1, -3) and C (4, -3, 1).

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Q 17Solve the Following Question.(5 Marks)5 Marks

Find a unit vector perpendicular to the plane containing the vectors $\vec{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}.}$

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Q 18Solve the Following Question.(5 Marks)5 Marks

Find all vectors of magnitude $10\sqrt{3}$ that are perpendicular to the plane of $\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}$ and $-\hat{\text{i}}+3\hat{\text{i}}+4\hat{\text{k}}.$

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Q 19Solve the Following Question.(5 Marks)5 Marks

If $\vec{\text{a}}=2\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}},\vec{\text{b}}=-3\hat{\text{i}}+4\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}},$ compute $\big(\vec{\text{a}}\times\vec{\text{b}}\big)\times\vec{\text{c}}$ and $\vec{\text{a}}\times\big(\vec{\text{b}}\times\vec{\text{c}}\big)$ and verify that these are not equal.

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Q 20Solve the Following Question.(5 Marks)5 Marks

If $\vec{\text{p}}$ and $\vec{\text{q}}$ are unit vectors forming an angle of 30°; find the area of the parallelogram having $\vec{\text{a}}=\vec{\text{p}}+2\vec{\text{q}}$ and $\vec{\text{b}}=2\vec{\text{p}}+\vec{\text{q}}$ as its diagonals.

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Vector or Cross Product question types

Vector or Cross Product questions

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Vector or Cross Product Maths - Vector or Cross Product

Vector or Cross Product question types

Vector or Cross Product questions

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