Find the unit vector that has the same direction as the vector v = 3i + j

Solution:

Given, the vector v is 3i - j

We need to find a unit vector that has the same direction as the given vector.

For the same direction, the sign configuration (+ - ) for the components remains unchanged.

For any vector v, the unit vector in the direction of v is (1/|v|) × v

Where, |v|= √(sum of squares of the magnitudes of the components of v)

|v| = √(3² + (-1)²)

|v| = √(9 + 1)

|v| = √10

Therefore, the unit vector is (3i - j)/√10.

Find the unit vector that has the same direction as the vector v = 3i + j

Summary:

A unit vector that has the same direction as the given vector v = 3i - j is (3i - j)/√10.