If, a, = 10, b, = 2 and a * b = 12, then value of, a x b, is : Option 1: 5 Option 2: 10 Option 3: 14 Option 4: 16

Answer: 16. Solution: Given, a, = 10, b, = 2, and a * b = 12. The dot product formula is: a * b =, a, b, The cross product formula is:, a x b, =, a, b, where is the angle between the two vectors. Using the dot product formula: 12 = 10 x 2 x 12 = 20 = 12/20 = 3/5 Using the identity ^2 + ^2 = 1: ^2 = 1 - ^2 ^2 = 1 - (3/5)^2 ^2 = 1 - 9/25 ^2 = 16/25 = 4/5 The angle between two vectors is always between 0° and 180°, so is positive. Using the cross product formula:, a x b, =, a, b, a x b, = 10 x 2 x 4/5, a x b, = 20 x 4/5, a x b, = 16 Therefore, a x b, = 16.

Algebra

Vector Algebra

Medium

core

If $|\vec{a}| = 10$ , $|\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$ , then value of $|\vec{a} \times \vec{b}|$ is :

✅ Correct Option: 4

Given $|\vec{a}| = 10$ , $|\vec{b}| = 2$ , and $\vec{a} \cdot \vec{b} = 12$ .

The dot product formula is:

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$

The cross product formula is:

$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$

where $\theta$ is the angle between the two vectors.

Using the dot product formula:

$12 = 10 \times 2 \times \cos\theta$

$12 = 20\cos\theta$

$\cos\theta = \dfrac{12}{20}$

$\cos\theta = \dfrac{3}{5}$

Using the identity $\sin^2\theta + \cos^2\theta = 1$ :

$\sin^2\theta = 1 - \cos^2\theta$

$\sin^2\theta = 1 - \left(\dfrac{3}{5}\right)^2$

$\sin^2\theta = 1 - \dfrac{9}{25}$

$\sin^2\theta = \dfrac{16}{25}$

$\sin\theta = \dfrac{4}{5}$

The angle between two vectors is always between $0°$ and $180°$ , so $\sin\theta$ is positive.

Using the cross product formula:

$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$

$|\vec{a} \times \vec{b}| = 10 \times 2 \times \dfrac{4}{5}$

$|\vec{a} \times \vec{b}| = 20 \times \dfrac{4}{5}$

$|\vec{a} \times \vec{b}| = 16$

Therefore, $|\vec{a} \times \vec{b}| = 16$ .

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The unit vector in the direction of the sum of vectors $\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$ is :

If ∣a⃗∣=10|\vec{a}| = 10∣a∣=10, ∣b⃗∣=2|\vec{b}| = 2∣b∣=2 and a⃗⋅b⃗=12\vec{a} \cdot \vec{b} = 12a⋅b=12, then value of ∣a⃗×b⃗∣|\vec{a} \times \vec{b}|∣a×b∣ is :

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If $|\vec{a}| = 10$ , $|\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$ , then value of $|\vec{a} \times \vec{b}|$ is :