If vector A = 6i + 8j, what is the unit vector in the direction of A? (2023)

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Question: If vector A = 6i + 8j, what is the unit vector in the direction of A? (2023)

Options:

Correct Answer: 3/5 i + 4/5 j

Exam Year: 2023

Solution:

Magnitude |A| = √(6^2 + 8^2) = 10. Unit vector = (1/10)(6i + 8j) = (3/5)i + (4/5)j.

If vector A = 6i + 8j, what is the unit vector in the direction of A? (2023)

If vector A = 6i + 8j, what is the unit vector in the direction of A? (2023)

Step 1: Identify the components of vector A. Here, A = 6i + 8j, which means the x-component is 6 and the y-component is 8.
Step 2: Calculate the magnitude of vector A using the formula |A| = √(x^2 + y^2). Substitute the values: |A| = √(6^2 + 8^2).
Step 3: Calculate 6^2 = 36 and 8^2 = 64. Then add them together: 36 + 64 = 100.
Step 4: Take the square root of 100 to find the magnitude: |A| = √100 = 10.
Step 5: To find the unit vector in the direction of A, use the formula: unit vector = (1/|A|) * A.
Step 6: Substitute the magnitude into the formula: unit vector = (1/10) * (6i + 8j).
Step 7: Distribute (1/10) to both components: unit vector = (6/10)i + (8/10)j.
Step 8: Simplify the fractions: (6/10) = (3/5) and (8/10) = (4/5).
Step 9: Write the final unit vector: unit vector = (3/5)i + (4/5)j.

Vector Magnitude – Understanding how to calculate the magnitude of a vector using the Pythagorean theorem.
Unit Vector – Knowing how to find a unit vector by dividing the vector by its magnitude.