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Numerade Educator

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Problem 25 Easy Difficulty

Find a unit vector that has the same direction as the given vector.

$ 8i - j + 4k $

Answer

$$
\frac{8}{9} \mathbf{i}-\frac{1}{9} \mathbf{j}+\frac{4}{9} \mathbf{k}
$$

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Video Transcript

let's find a unit vector that has the same direction as a vector. Eight I minus shapeless four K. So remember that given a vector in space any multiple or scalable multiple of that picture have the same direction as to give him victor. So what we need to find here is that magnitude of this factor and multiply this given vector by the Scaler one over the magnitude of right. The uh of these factors. The magnitude we have found that is because if we have a victor V and we find W equals one over the norm of B or the magnitude sci fi. Yeah, this is a scalar, a real number and this is a given vector in space. For example, then if we calculate the norm of this new vector, W we know is equal to the absolute value of the Scaler times. Mm Yeah. The norm of the victor does the property of the norm. That is if we have skater and vector and we are calculating the norm or magnitude of that victory, we know that is equal to the absolute value of the skater at times the norm of the victor. So here, because we have W equals scalar times vector. We apply this rule here and we get that the norm W Is the absolute value of the scaler one over norm of B times the norm of this vector. There is a normal fee but this scalar is positive because the norm is always positive or no number. So here we have to ask that the victor, W is not zero in order to calculate that as one over the bandages of w. And because it's positive, is this the same value? That is one over the north of w. Uh fi sorry I meant because here we have P as we can see. So here is B. And here is we times these enormity And this is clearly one number over itself is one. That is if we have any non null vector, we always can construct a vector, a unit vector in this with the same direction as this vector and that is one over the magnitude of the given vector times the victor. This is the scaler we gotta use and in this case we obtain a vector with the same direction after given vector and With nor or magnitude equal to one. So after recording all these facts, we know then that the given vectors, we call it v eight, I -J plus four K. Then to vector W equals one over the norm of fee. That's right. Just like the other V, the norm of V. We are talking about the even victor here, times V is a new victor. That has the same direction as V because being a multi scalar multiple multiple fee, it lies on the same right, a straight line. So it has the same direction as V N. Its norm is magnitude is he's one. That is it a unit Mhm factor. So what we can find is first the norm of fee is the square root of this, sum of the square of the components of the That is eight square Plus -1 sq plus four square. As we can see the coordinates of the vector. V r eight negative one and four. So this is the square root of 64 Plus one plus 16 and 64. 16. We get 10, 80 81. Spirit of 81. And that is nine. Perfect. Mhm. So this is the magnitude or norm of the given vector. Then the yearning victor, dad has the same. Okay, direction us. The given vector is W equals one over the normal fee times feed, that is 1/9 times fee. That is 1/9 times eight. I minus J plus for K. Or what is the same? Is 8/9 I minus one. Ninth G Plus 4, 9th K. So this is the victor we are looking for with the same direction as the given vector and is a unit vector. And we verified if you wanted to see in a vector simply by calculating the norm or magnitude of w is the square root of the sum of the square of its components. That is 8/9 square plus negative 1/9 square plus 4/9 Square. And that is the square root off 64/9 square. I'm gonna Stick with this. nine square instead of calculating it because it's going to be easier than is one overnight square plus 16. Okay, 16 overnight square. Which And because all the denominators are nine square, we can simply some distraction ist maintaining that common denominator and some the numerator. So we get 64 plus one plus 16. And this operation in the numerator, we have done it before is 81. In fact, and now we know nine square is 81, so we get Spirit of one is 1. So we found that w is uh unit factor. And the fact that it is, it was obtained as a scalar times the given vector. It has the same direction and that completes the exercise.