Question

\(|\vec{a}\times\vec{b}|^2+|\vec{a}.\vec{b}|^2=144\) and \(|\vec{a}|=4\) then \(|\vec{b}|\) is equal to

• A. 8
• B. 12
• C. 4
• D. 3

Hint:

When given the sum of squares of the cross product and dot product of two vectors, use the identity \( |\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2 \) to simplify the equation. Then, solve for the unknown magnitude of the second vector.

Answer 1

The Correct Option is D

The correct answer is: (D): 4

We are given the following information:

|\vec{a} \times \vec{b}|^2 + |\vec{a} \cdot \vec{b}|^2 = 144

|\vec{a}| = 4

We are tasked with finding \( |\vec{b}| \).

Step 1: Use the identity for cross and dot products

We know the following identity for vectors \( \vec{a} \) and \( \vec{b} \):

|\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2

Substitute this into the given equation:

(|\vec{a}|^2 |\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2) + |\vec{a} \cdot \vec{b}|^2 = 144

Step 2: Simplify the equation

Notice that the second term and the last term involve \( (\vec{a} \cdot \vec{b})^2 \), so they cancel out. This simplifies to:

|\vec{a}|^2 |\vec{b}|^2 = 144

Step 3: Substitute the known value for \( |\vec{a}| \)

We are given that \( |\vec{a}| = 4 \), so substitute this into the equation:

4^2 |\vec{b}|^2 = 144

Step 4: Solve for \( |\vec{b}| \)

Now, simplify and solve for \( |\vec{b}| \):

16 |\vec{b}|^2 = 144

|\vec{b}|^2 = \frac{144}{16} = 9

Taking the square root of both sides gives:

|\vec{b}| = 3

Conclusion:
The correct value of \( |\vec{b}| \) is \( 4 \), so the correct answer is (D): \( 4 \).