Question

If |\(\vec{a}\)| = 5, |\(\vec{b}\)| = 2 and |\(\vec{a}\) · \(\vec{b}\) | = 8 then the value of |\(\vec{a} \)×\(\vec{b} \)| is:

• A. 5
• B. 6
• C. 36
• D. ±6

Answer 1

The Correct Option is B

To find the magnitude of the cross product |\(\vec{a} \times \vec{b}\)|, we use the formula:
|\(\vec{a} \times \vec{b}\)| = |\(\vec{a}\)| |\(\vec{b}\)| sin\(\theta\)

We also have the dot product formula:
|\(\vec{a} \cdot \vec{b}\)| = |\(\vec{a}\)| |\(\vec{b}\)| cos\(\theta\)

Given, |\(\vec{a}\)| = 5, |\(\vec{b}\)| = 2, and |\(\vec{a} \cdot \vec{b}\)| = 8.

Using the dot product formula:
8 = 5 × 2 × cos\(\theta\)
cos\(\theta\) = \(\frac{8}{10} = \frac{4}{5}\)

Now, to find sin\(\theta\), we use the Pythagorean identity:
sin\(\theta\) = √(1 - cos²\(\theta\))
sin\(\theta\) = √(1 - \(\left(\frac{4}{5}\right)^2 = \frac{16}{25}\))
sin\(\theta\) = √(\(\frac{25 - 16}{25} = \frac{9}{25}\))
sin\(\theta\) = √\(\frac{9}{25}\) = \(\frac{3}{5}\)

Finally, compute the magnitude of the cross product:
|\(\vec{a} \times \vec{b}\)| = 5 × 2 × \(\frac{3}{5}\) = 6

Therefore, the value of |\(\vec{a} \times \vec{b}\)| is 6.