Find the unit vector in the direction of the vector v = (4, -3).

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Question: Find the unit vector in the direction of the vector v = (4, -3).

Options:

Correct Answer: (4/5, -3/5)

Solution:

Magnitude |v| = √(4^2 + (-3)^2) = √(16 + 9) = 5. Unit vector = (4/5, -3/5).

Find the unit vector in the direction of the vector v = (4, -3).

Find the unit vector in the direction of the vector v = (4, -3).

Correct Answer: (4/5, -3/5)

Step 1: Identify the vector v, which is (4, -3).
Step 2: Calculate the magnitude of the vector v using the formula |v| = √(x^2 + y^2), where x and y are the components of the vector.
Step 3: Substitute the values into the formula: |v| = √(4^2 + (-3)^2).
Step 4: Calculate 4^2, which is 16, and (-3)^2, which is 9.
Step 5: Add the results: 16 + 9 = 25.
Step 6: Take the square root of 25 to find the magnitude: √25 = 5.
Step 7: To find the unit vector, divide each component of the vector v by its magnitude.
Step 8: Divide the first component: 4 / 5.
Step 9: Divide the second component: -3 / 5.
Step 10: Write the unit vector as (4/5, -3/5).

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 each component of the vector by its magnitude.