Find a unit vector in the direction of the given vector v = (1, 3).
Vectors are quantities with both magnitude and direction. Vectors are used to represent quantities in physics, such as velocity, acceleration, force, or torque.
Answer: The unit vector for v = (1, 3) will be (1/√10, 3/√10).
A unit vector is a vector with a magnitude of 1. It is also known as a direction vector.
For example, vector v = (2, 4) is not a unit vector, because its magnitude is not equal to 1, i.e., |v| = √(22 + 42) ≠ 1. Any vector can become a unit vector by dividing it by the magnitude of the given vector.
To calculate the unit vector for v = (1, 3) we need to divide the vector by its magnitude.
Magnitude of the vector v = (1, 3) is:
|v| = √(12 + 32) = √(10)
Hence unit vector is (1, 3)/√10 or (1/√10, 3/√10)
Thus, the unit vector for v = (1, 3) will be (1/√10, 3/√10).