If a vector of magnitude 2 units makes an angle π/3 with 2î, π/4 with 3ĵ and an acute angle θ with 4k̂, then what are the components of the vector?

Let the vector be (x, y, z) with magnitude 2. The angle with 2î (i.e., with î direction) is π/3, so cos(π/3) = x/2 = 1/2, giving x = 1. Angle with 3ĵ is π/4, so cos(π/4) = y/2 = 1/√2, giving y = 2/√2 = √2. Magnitude: x² + y² + z² = 4 ⟹ 1 + 2 + z² = 4 ⟹ z² = 1 ⟹ z = ±1. For acute angle with k̂, cosθ = z/2 > 0, so z = 1. Components: (1, √2, 1).