# Question Video: Finding the Polar Form of a Vector Mathematics

Complete the following: If 𝐀 = 〈2, 10°〉, then the polar form of vector 8𝐀 is ＿.

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### 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.