Find the angle between the vectors (1, 0, 0) and (0, 1, 0).

Find the angle between the vectors (1, 0, 0) and (0, 1, 0).

Correct Answer: 90 degrees

Step 1: Identify the two vectors. The first vector is u = (1, 0, 0) and the second vector is v = (0, 1, 0).
Step 2: Calculate the dot product of the two vectors, u · v. This is done by multiplying the corresponding components: (1 * 0) + (0 * 1) + (0 * 0) = 0.
Step 3: Calculate the magnitude (length) of each vector. For vector u, |u| = √(1^2 + 0^2 + 0^2) = √1 = 1. For vector v, |v| = √(0^2 + 1^2 + 0^2) = √1 = 1.
Step 4: Use the formula for the angle θ between two vectors: θ = cos⁻¹((u · v) / (|u| |v|)). Substitute the values: θ = cos⁻¹(0 / (1 * 1)) = cos⁻¹(0).
Step 5: Find the angle whose cosine is 0. This angle is 90 degrees.

Dot Product – The dot product of two vectors is used to find the cosine of the angle between them.
Magnitude of Vectors – The magnitude of a vector is calculated using the square root of the sum of the squares of its components.
Angle Between Vectors – The angle between two vectors can be determined using the inverse cosine of the dot product divided by the product of their magnitudes.