Question

If $\vec{u}=\hat{i}-3\hat{j}+2\hat{k}$ and $\vec{v}=2\hat{i}+4\hat{j}-5\hat{k}$, then $|\vec{u}\times\vec{v}|^{2}+|\vec{u}\cdot\vec{v}|^{2}=$

• A. 640
• B. 630
• C. 690
• D. 740
• E. 730

Hint:

Vector Tip: Lagrange's Identity is a massive time-saver! Never manually calculate the determinant for the cross product if a question only asks for the sum of the squared cross and dot products.

Answer 1

The Correct Option is B

Concept:
This problem can be solved instantly using Lagrange's Identity, which relates the cross product and dot product of two vectors directly to their individual magnitudes without needing to compute the cross product vector itself: $$|\vec{u} \times \vec{v}|^2 + |\vec{u} \cdot \vec{v}|^2 = |\vec{u}|^2 |\vec{v}|^2$$

Step 1: State the given vectors.
We are given the coordinate vectors: $$\vec{u} = \hat{i} - 3\hat{j} + 2\hat{k}$$ $$\vec{v} = 2\hat{i} + 4\hat{j} - 5\hat{k}$$

Step 2: Calculate the squared magnitude of u.
The squared magnitude of a vector is the sum of the squares of its scalar components: $$|\vec{u}|^2 = (1)^2 + (-3)^2 + (2)^2$$ $$|\vec{u}|^2 = 1 + 9 + 4 = 14$$

Step 3: Calculate the squared magnitude of v.
Similarly, find the squared magnitude for the second vector: $$|\vec{v}|^2 = (2)^2 + (4)^2 + (-5)^2$$ $$|\vec{v}|^2 = 4 + 16 + 25 = 45$$

Step 4: Apply Lagrange's Identity.
Substitute the scalar squared magnitudes into the identity: $$|\vec{u} \times \vec{v}|^2 + |\vec{u} \cdot \vec{v}|^2 = (14)(45)$$

Step 5: Compute the final product.
Multiply the two integer values to obtain the final result: $$14 \times 45 = 630$$ Hence the correct answer is (B) 630.