Taking a → = 2 i ^ - 3 j ^ - k ^ and b → = i ^ + 4 j ^ - 2 k ^ , verify that a → × b → ≠ b → × a → .

Given a → = 2 i ^ - 3 j ^ - k ^ and b → = i ^ + 4 j ^ - 2 k ^ , Now, a → × b → = i ^ j ^ k ^ 2 - 3 - 1 1 4 - 2 = 6 + 4 i ^ - - 4 + 1 j ^ + 8 + 3 k ^ = 10 i ^ + 3 j ^ + 11 k ^ And, b → × a → = i ^ j ^ k ^ 1 4 - 2 2 - 3 - 1 = - 4 - 6 i ^ - - 1 + 4 j ^ + - 3 - 8 k ^ = - 10 i ^ - 3 j ^ - 11 k ^ ⇒ a → × b → = - b → × a → , but a → × b → ≠ b → × a → . Hence proved.

If a → = 2 i ^ - j ^ + k ^ , b → = i ^ + 2 j ^ - k ^ and c → = 2 i ^ + 3 j ^ , find a → × b → × c → and a → × b → × c → , and verify that these are not the same.

Given a → = 2 i ^ - j ^ + k ^ , b → = i ^ + 2 j ^ - k ^ and c → = 2 i ^ + 3 j ^ . Now, a → × b → = i ^ j ^ k ^ 2 - 1 1 1 2 - 1 = 1 - 2 i ^ - - 2 - 1 j ^ + 4 + 1 k ^ = - i ^ + 3 j ^ + 5 k ^ ⇒ a → × b → × c → = i ^ j ^ k ^ - 1 3 5 2 3 0 = 0 - 15 i ^ - 0 - 10 j ^ + - 3 - 6 k ^ = - 15 i ^ + 10 j ^ - 9 k ^ And, b → × c → = i ^ j ^ k ^ 1 2 - 1 2 3 0 = 0 + 3 i ^ - 0 + 2 j ^ + 3 - 4 k ^ = 3 i ^ - 2 j ^ - k ^ ⇒ a → × b → × c → = i ^ j ^ k ^ 2 - 1 1 3 - 2 - 1 = 1 + 2 i ^ - - 2 - 3 j ^ + - 4 + 3 k ^ = 3 i ^ + 5 j ^ - k ^ Hence, a → × b → × c → and a → × b → × c → are not the same.

If a → = 3 i ^ + j ^ + 2 k ^ and b → = 2 i ^ - 2 j ^ + 4 k ^ , then find a unit vector perpendicular to both the vectors a → and b → .

Given a → = 3 i ^ + j ^ + 2 k ^ & b → = 2 i ^ - 2 j ^ + 4 k ^ , a → × b → = i ^ j ^ k ^ 3 1 2 2 - 2 4 = 4 + 4 i ^ - 12 - 4 j ^ + - 6 - 2 k ^ = 8 i ^ - 8 j ^ - 8 k ^ ∴ Magnitude of a → × b → = a → × b → = 8 2 + - 8 2 + - 8 2 = 8 3 n ^ unit vector perpendicular to both a → and b → = a → × b → a → × b → = 8 i ^ - 8 j ^ - 8 k ^ 8 3 = 1 3 i ^ - j ^ - k ^ There are two unit vectors perpendicular to both a → and b → . They are ± n ^ = ± 1 3 i ^ - j ^ - k ^ .

E M B I B E

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>EXERCISE>Q 13

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IMPORTANT

Earn 100

If $\vec{a} = 4 \hat{i} + 3 \hat{j} + 2 \hat{k}$ and $\vec{b} = 3 \hat{i} + 2 \hat{k}$ , then find $|\vec{b} \times 2 \vec{a}|$ .

50% studentsanswered this correctly

Important Questions on Vectors

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>EXERCISE>Q 13

If

\vec{a} = \hat{i} + \hat{j} - 3 \hat{k}

and

\vec{b} = \hat{j} + 2 \hat{k}

, then find the value of

k

, when

|2 \vec{b} \times \vec{a}| = 2 \sqrt{k}

.

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>EXERCISE>Q 14

If $\vec{a} = 3 \hat{i} - \hat{j} - 2 \hat{k}$ and $\vec{b} = 2 \hat{i} + 3 \hat{j} + \hat{k}$ , then $(\vec{a} + 2 \vec{b}) \times (2 \vec{a} - \vec{b}) = k \hat{i} + 35 \hat{j} - 55 \hat{k}$ , hence, find the value of $k$ .

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>EXERCISE>Q 14

Taking

\vec{a} = 2 \hat{i} - 3 \hat{j} - \hat{k}

and

\vec{b} = \hat{i} + 4 \hat{j} - 2 \hat{k}

, verify that

\vec{a} \times \vec{b} \neq \vec{b} \times \vec{a}

.

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>EXERCISE>Q 15

If $\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}$ , $\vec{b} = \hat{i} + 2 \hat{j} - \hat{k}$ and $\vec{c} = 2 \hat{i} + 3 \hat{j}$ , find $(\vec{a} \times \vec{b}) \times \vec{c}$ and $\vec{a} \times (\vec{b} \times \vec{c})$ , and verify that these are not the same.

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>EXERCISE>Q 15

If $\vec{a}, \vec{b}$ and $\vec{c}$ represent the position vectors of the points with coordinates $(2, - 10, 2), (3, 1, 2)$ and $(2, 1, 3)$ respectively, find the value of $\vec{a} \times (\vec{b} \times \vec{c}) .$

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>EXERCISE>Q 16

If

θ

is the angle between two vectors

\hat{i} - 2 \hat{j} + 3 \hat{k}

and

3 \hat{i} - 2 \hat{j} + \hat{k}

, find the value of

k

, if

\sin (θ) = \frac{2 \sqrt{k}}{7}

.

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>EXERCISE>Q 17

If $\vec{a} = 3 \hat{i} + \hat{j} + 2 \hat{k}$ and $\vec{b} = 2 \hat{i} - 2 \hat{j} + 4 \hat{k}$ , then find the value of $k$ if sine of the angle between $\vec{a}$ and $\vec{b}$ is $\frac{k}{\sqrt{7}}$

MEDIUM

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 1 - Vectors>EXERCISE>Q 17

If $\vec{a} = 3 \hat{i} + \hat{j} + 2 \hat{k}$ and $\vec{b} = 2 \hat{i} - 2 \hat{j} + 4 \hat{k}$ , then find a unit vector perpendicular to both the vectors $\vec{a}$ and $\vec{b}$ .