00:01
We are given the two vectors a vector and b vector and we have to determine if these two vectors are orthogonal, parallel or neither.
00:09
So we see that two vectors are orthogonal when their dot product is zero.
00:14
That is when a dot b and this equals zero, we see that the two vectors are orthogonal.
00:24
And we see that the two vectors are parallel.
00:27
Well, if one vector can be written as a scalar multiple of another vector, that is, if we can write a equals lambda b, or if we can write b equals some t times of a, where lambda and t are some scalars, then we say that these two vectors are parallel.
00:51
The shortest way to determine the parallel vectors are we find the ratio of the corresponding components.
00:59
And if the ratio is the same we then say that the two vectors are parallel ratio of the components are equal to each other okay so let's see how to determine this orthogonal parallel or neither first i'm going to do the dot product so let me do the dot product that is a dot b and this equals we have to find the multiples of the corresponding components so we have negative 8 for i and 6 for g so we have to multiply these two that is negative 8 times 6 then we add with the other multiple that is 12 with negative 9 that is for j component that is 12 times of negative 9 and then we add with this the multiple of k component that is k component is 4 and here it is negative 3 so we do 4 times of negative 3 let's check if it is giving as 0.
02:08
This is negative 48 and this is negative 108.
02:13
4 times the negative 3 is negative 12.
02:16
And so this gives negative 168.
02:25
So, negative 168.
02:27
And clearly this is not equal to 0.
02:30
So since we did not get a dot be equal to 0 we can say that this is not orthogonal and this means this could be either parallel or neither so now let's determine if it is parallel so for this i'm going to take the corresponding components and find the ratio of the corresponding components so first let me write the a vector in component form so a vector is negative 8 12 and 4 this is the component form of a vector.
03:11
Also let me write the component form of v vector that is 6, negative 9 and negative 3.
03:20
And let's find the ratio of the corresponding components.
03:23
So here we take this x component is negative 8 and this is 6 for b vector.
03:30
So we find the ratio that is negative 8 by 6.
03:33
And this we have to check if this is equal to the other ratio of this y component.
03:40
Is 12 by negative 9.
03:42
So this is 12 by negative 9 and we should also check if this equals the last component that is 4 by negative 3.
03:55
Now let's check this.
03:57
We can simplify this.
03:59
This is 4 times 2.
04:01
This is 3 times 2 and here also this is 4 times 3 is 12 and this is 3 times negative 3 times 3 is negative so we can see that the ratio finally reduces to negative 4 thirds and here also we have negative 4 thirds and here also we have negative 4 thirds.
04:25
So we check that the ratio of the corresponding components are equal to each other which means the vectors a vector and b vector are parallel a and b vectors are parallel.
04:48
It's to part b.
04:50
So here we have the vectors a vector equals 3i minus j plus 3k and b vector equals 5i plus 9 j minus 2k.
04:59
So first let me do the dot product that is a dot b and this equals i have to find the product of the corresponding components.
05:07
So i do three times of 5 plus the j component is negative 1 here and it is 9 here.
05:19
So i multiplied these two plus the k component is 3.
05:24
Multiply with the k component is negative 2 here.
05:28
So let's check this.
05:29
This is 3 times 5 is 15 and negative 1 times positive 9 is negative 9.
05:36
And this is 3 times negative 2 is negative 6.
05:39
So this gives 15 minus negative 9, negative 6 is negative 15.
05:45
So 15 minus 15 equals 0.
05:47
So clearly we see that a dot b equals 0 and this means these two vectors, that is a vector and b vector or orthogonal to each other...