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Checking and Constructing Orthogonal or Parallel Vectors

Determine whether the given vectors are orthogonal, parallel or neither.
<-5,3,7> and <6,-1,2>
<4,6> and <-3,2>
-i + 2j + 5k and 3i + 4j - k
2i + 6j - 4k and -3i -9j +6k

Find a unit vector that is orthogonal to both i+j and i+k.

If a = <3,0,-1>, find a vector b such that comp_a b = 2 (component of b in the a direction is 2).


Solution Preview

Please see the attached file. Included in text here as much as possible, but the file is better.
In this answer, I'm generally going to change i, j, and k to the appropriate numerical vectors, because it's simpler this way. However, you could do those problems without that, and most likely you'll have to as things go on in the course.

a. We check for orthogonality by taking the dot product of the two vectors. If it's 0, they're orthogonal. Remember that the dot product is the sum of the products of the appropriate component pairs.
a?b = -5*6 + 3*(-8) + 7*2 = -30 - 24 +14 = -40 They are not orthogonal
We check for parallelness by checking if the vectors are scalar multiples of each other (that is, you can multiply one by a specific number to get the other). We can do this by taking pairs of components (for example the -5 and 6 pair) and dividing one by the other, and seeing if we always get the same answer.
-5/6, 3/(-8), and 7/2 : are they the same number? No, certainly not; for example -5/6 is not the same as 7/2.
a?b = 4*(-3) + 6*2 = -12 + 12 = 0
They are orthogonal! This also means that they are not parallel; nonzero orthogonal vectors cannot also be parallel. Note that 4/(-3) is not the same as 6/2.
c. Here I will rewrite the unit vector sums as numerical vectors.
a = -1*i + 2*j + 5*k = -1*<1, 0, 0> + 2*<0, 1, 0> + 5*<0, 0, 1> = <-1, 2, 5>
b = <3, 4, -1>
a?b = -1*3 + 2*4 + 5*(-1) = -3 + 8 -5 = 0
Again the vectors are orthogonal, and therefore not parallel. Note that -1/3 is not equal ...

Solution Summary

This solution shows how to check four different pairs of vectors to see if they are orthogonal, parallel, or neither. It also shows how to construct a unit vector orthogonal to two other vectors, and how to construct a vector with a given component in a given direction.