Video Transcript
Complete the following. If vector 𝐀 has magnitude two and
angle 10 degrees, then the polar form of vector eight 𝐀 is what.
Remember, a vector is a way of
describing the magnitude, or size, and direction of a path, denoted by a line
segment. When we describe a vector using
polar form, we use the magnitude, 𝑟, which essentially means the length of the path
from the start point to endpoint, and an angle 𝜃. In particular, this is the angle
that the line segment makes with the positive 𝑥-axis in a counterclockwise
direction. This means that the vector 𝐀 has a
length of two units and makes an angle of 10 degrees with the positive 𝑥-axis in a
counterclockwise direction. We can represent this on the
Cartesian plane by assuming it starts at the origin as shown.
So, with this in mind, let’s
represent the vector eight 𝐀 in polar form. Remember, when we multiply a vector
by a scalar, this changes the length of the line segment. We can represent this
diagrammatically as shown. For the vector eight 𝐀, the line
segment is eight times the length of vector 𝐀. The direction remains
unchanged. In fact, we can say that the angle
that the vector eight 𝐀 makes with the positive real axis is 10 degrees, as in the
case of the original vector. This means we can write the vector
eight 𝐀 in polar form by simply multiplying the magnitude of vector 𝐀 by eight and
leaving the angle unchanged. Hence, the polar form of the vector
eight 𝐀 is 16, 10 degrees.
