Question Video: Finding the Polar Form of a Vector | Nagwa Question Video: Finding the Polar Form of a Vector | Nagwa

Question Video: Finding the Polar Form of a Vector Mathematics • First Year of Secondary School

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Complete the following: If the vector form of πŽπ€ is 2𝐒 + 2𝐣, then the polar form of vector π€πŽ is οΌΏ.

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Video Transcript

Complete the following. If the vector form of vector πŽπ€ is two 𝐒 plus two 𝐣, then the polar form of vector π€πŽ is what.

We should first note that the vector we’re looking to find in polar form is not the vector we’ve been given in vector form. We’ve been given vector πŽπ€, and we’re looking for vector π€πŽ. Before we can find the polar form of vector π€πŽ, we’ll first need to write it down in vector form. Vector π€πŽ can be seen as the reverse of vector πŽπ€. It will have the same magnitude, but the opposite direction. So, given that vector πŽπ€ is two 𝐒 plus two 𝐣, then vector π€πŽ will be negative two 𝐒 plus negative two 𝐣, which we can write more simply as negative two 𝐒 minus two 𝐣.

Now, we need to express this in polar form. The vector form expressed π€πŽ in terms of the displacements in the π‘₯- and 𝑦-directions, but the polar form expresses it in terms of the vector’s magnitude π‘Ÿ and its angle πœƒ. Let’s sketch a diagram to represent vector π€πŽ. The vector negative two 𝐒 minus two 𝐣 represents a movement of negative two in the π‘₯-direction and negative two in the 𝑦-direction. If we position the vector to start at the origin, we see that it lies in the third quadrant. The length of this vector is given by π‘Ÿ, and the angle πœƒ is measured in the counterclockwise direction from the positive π‘₯-axis.

The value of π‘Ÿ can be calculated using the Pythagorean theorem as the square root of π‘₯ squared plus 𝑦 squared. And the tangent of angle πœƒ can be represented as 𝑦 divided by π‘₯. For vector π€πŽ, we can see that the change in π‘₯-value is negative two, and the change in the 𝑦-value is also negative two. To calculate the magnitude of the vector, we need to calculate the square root of negative two squared plus negative two squared. That’s the square root of four plus four, which simplifies to two root two.

Now, tan πœƒ is equal to the 𝑦-value over the π‘₯-value, which is negative two over negative two, which simplifies to one. This means that we need to find the inverse tan of one to calculate the value of πœƒ. Our calculator gives the result that πœƒ is equal to 45 degrees. However, if we look back at our sketch, we know this isn’t correct; πœƒ is much greater than 45 degrees.

The calculator actually gave us the measure of the acute angle between our vector and the π‘₯-axis, marked here in green. It happens to have the same value as its tangent. So, we need to add this extra 180 degrees to find the true value of πœƒ as the measure of the counterclockwise angle between the positive π‘₯-axis and our vector π€πŽ. And that gives us 225 degrees.

It’s always a good idea to sketch the vector starting from the origin on a pair of coordinate axes, as we did, before finding the angle for the polar form, just to make sure we correctly take account of which quadrant it lies in. So, the answer to the question is that, in polar form, the vector π€πŽ has magnitude two root two and an angle of 225 degrees.

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