If vector A = 5i + 12j and vector B = -5i + 12j, what is the angle between them?

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Question: If vector A = 5i + 12j and vector B = -5i + 12j, what is the angle between them?

Options:

Correct Answer: 180 degrees

Solution:

A · B = (5)(-5) + (12)(12) = -25 + 144 = 119. Since A and B are in opposite directions, the angle is 180 degrees.

If vector A = 5i + 12j and vector B = -5i + 12j, what is the angle between them?

If vector A = 5i + 12j and vector B = -5i + 12j, what is the angle between them?

Step 1: Identify the components of vector A, which are 5i and 12j.
Step 2: Identify the components of vector B, which are -5i and 12j.
Step 3: Calculate the dot product of vectors A and B using the formula A · B = (A_x * B_x) + (A_y * B_y).
Step 4: Substitute the values: A · B = (5 * -5) + (12 * 12).
Step 5: Calculate the products: -25 + 144.
Step 6: Add the results: -25 + 144 = 119.
Step 7: Determine the direction of the vectors. Vector A points in the positive x direction and vector B points in the negative x direction, but both point in the positive y direction.
Step 8: Since vector A and vector B are in opposite directions along the x-axis, the angle between them is 180 degrees.

Dot Product – The dot product of two vectors is used to find the cosine of the angle between them.
Vector Components – Understanding how to break down vectors into their i and j components is crucial for calculations.
Angle Calculation – The angle between two vectors can be calculated using the dot product and the magnitudes of the vectors.