Given two displacement vectors A⃗ = (3.00iˆ-4.00jˆ+4.00kˆ) mA→ = (3.00i^-4.00j^+4.00k^) m and B⃗ = (2.00iˆ+3.00jˆ-7.00kˆ) mB→ = (2.00i^+3.00j^-7.00k^) m, find the displacements and their magnitudes for (a) C⃗ = A⃗ + B⃗ C→=A→+B→ and (b) D⃗ = 2A⃗ - B⃗ D→=2A→-B→.

Displacement C = (5.00iˆ - 1.00jˆ - 3.00kˆ) m

Magnitude = 5.92 m

Displacement D = (4.00iˆ-11.00jˆ+15.00kˆ) m

Magnitude = 19.03 m

Explanation:

Vector A = (3.00iˆ-4.00jˆ+4.00kˆ) m

Vector B = (2.00iˆ+3.00jˆ-7.00kˆ) m

a) Vector C = A + B = (3.00iˆ-4.00jˆ+4.00kˆ) + (2.00iˆ+3.00jˆ-7.00kˆ)

Vector addition is done component by component, that is, do î component, then j component and k component

C = (5.00iˆ - 1.00jˆ - 3.00kˆ) m

Magnitude of C = √[ (5²) + (-1) ² + (-3) ²] = √ (35) = 5.92 m

b) Vector D = 2A - B

D = 2 (3.00iˆ-4.00jˆ+4.00kˆ) - (2.00iˆ+3.00jˆ-7.00kˆ) = (6.00iˆ-8.00jˆ+8.00kˆ) - (2.00iˆ+3.00jˆ-7.00kˆ) = (4.00iˆ-11.00jˆ+15.00kˆ)

Magnitude of D = √[ (4²) + (-11) ² + (15) ²] = √ (362) = 19.03 m