If the vector a = (2, 2) and b = (2, -2), what is the angle between them?

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Question: If the vector a = (2, 2) and b = (2, -2), what is the angle between them?

Options:

Correct Answer: 90 degrees

Solution:

Angle = cos⁻¹((a·b) / (|a||b|)) = cos⁻¹(0) = 90 degrees

If the vector a = (2, 2) and b = (2, -2), what is the angle between them?

If the vector a = (2, 2) and b = (2, -2), what is the angle between them?

Step 1: Identify the vectors. We have vector a = (2, 2) and vector b = (2, -2).
Step 2: Calculate the dot product of vectors a and b. The dot product a·b = (2 * 2) + (2 * -2) = 4 - 4 = 0.
Step 3: Calculate the magnitude (length) of vector a. |a| = √(2^2 + 2^2) = √(4 + 4) = √8 = 2√2.
Step 4: Calculate the magnitude (length) of vector b. |b| = √(2^2 + (-2)^2) = √(4 + 4) = √8 = 2√2.
Step 5: Use the formula for the angle between two vectors: Angle = cos⁻¹((a·b) / (|a||b|)).
Step 6: Substitute the values into the formula: Angle = cos⁻¹(0 / (2√2 * 2√2)) = cos⁻¹(0 / 8) = cos⁻¹(0).
Step 7: Find the angle whose cosine is 0. The 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 formula |a| = √(x² + y²).
Inverse Cosine Function – The inverse cosine function (cos⁻¹) is used to determine the angle from the cosine value.