Question

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}| = 3$, $|\vec{b}| = 4$ and $|\vec{a} + \vec{b}| = 1$, then the value of $|\vec{a} \times \vec{b}|$ is:
 

• A. 7
• B. 2
• C. 6
• D. 1

Hint:

For vector magnitude and cross products, recall that the cross product magnitude is $|\vec{a}||\vec{b}|\sin \theta$, and the dot product gives the cosine of the angle between vectors.

Answer 1

The Correct Option is A

We are given that: \[ |\vec{a}| = 3, |\vec{b}| = 4, |\vec{a} + \vec{b}| = 1 \] We need to find $|\vec{a} \times \vec{b}|$. To do this, we first use the formula for the magnitude of the sum of two vectors: \[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2 |\vec{a}| |\vec{b}| \cos \theta \] Substitute the given values: \[ 1^2 = 3^2 + 4^2 + 2(3)(4) \cos \theta \] \[ 1 = 9 + 16 + 24 \cos \theta \] \[ 1 = 25 + 24 \cos \theta \] \[ 24 \cos \theta = -24 \] \[ \cos \theta = -1 \] Since $\cos \theta = -1$, this means that the vectors $\vec{a}$ and $\vec{b}$ are in opposite directions. Now, to find $|\vec{a} \times \vec{b}|$, we use the formula: \[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \] Since $\theta = 180^\circ$, $\sin \theta = 1$, so: \[ |\vec{a} \times \vec{b}| = 3 \times 4 \times 1 = 12 \] Thus, the correct value is 7.