1. Consider the following vectors and answer the...

1. Consider the following vectors and answer the following: vec{A} = vec{i} + 2vec{j} - 3vec{k}, quad vec{B} = 3vec{i} - vec{j} + 2vec{k}, (a) Show that, the vectors vec{C} = vec{A} + vec{B} and vec{D} = vec{A} - vec{B} are perpendicular to each other. (b) Find a unit vector vec{E} which is perpendicular to both vec{C} and vec{D}. (c) Find the angle between the vectors vec{A} and vec{B}. (d) Find the area of the parallelogram formed by the vectors vec{A} and vec{B}.

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00:01 In this problem we are given two vectors a and b and we are going to answer some questions about them.

00:08 Okay a is equal to 1, 2, minus 3 in this tuple notation and b is given by 3, minus 1 and 2.

00:20 Okay part a.

00:23 We define two vectors c and d as c equal to a plus b and d equal to a minus b.

00:34 We will show that c and d are perpendicular to each other.

00:39 Okay c can be found to be 4, 1, minus 1 and d is equal to minus 2, 3, minus 5.

00:52 So perpendicular means their dot product should be equal to 0.

00:57 So we have minus 8 plus 3 plus 5 which is equal to 0.

01:03 Therefore we say that c is perpendicular to d.

01:09 In the next part we are going to find a unit vector e which is perpendicular to both c and d.

01:25 Okay let's start with this general e vector and then we can normalize it later.

01:32 So i am going to indicate the components by e x, e y and e z.

01:38 So from the perpendicular relation with c we have e dot c equal to 0 and we also have e dot d equal to 0 with a similar reason.

01:55 Okay e dot c is 4, 4 e x, not exponential, plus e y minus e z.

02:06 And for the second dot product we have minus 2 e x plus 3 e y minus 5 e z.

02:15 From the first relation we can obtain e z equal to 4 e x plus e y and with that the second equation gives minus 11 e x.

02:31 So with that we obtain e z equal to minus 7 e x.

02:38 So we have this one free parameter to choose...

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