Which of the following are orthogonal pairs? a) 5i+3j b) i+j c) 3i-5j d) -7i+j?

Question

Please also tell me how you solved this question. Thanks.

Answer 1

a and c

Answer 2

It's pretty easy to identify, especially here with 2D vector spaces.
The answer is a) and c)
How do I know?
Well the vectors basically need to be perpendicular, so when the dot product of two vectors a and b is defined as /a/./b/ cos x
where x is the angle between the vectors. So when perpendicular x= pi/2 rad.
then a.b = 0
so the only pair here where this is true(by summation/matrix multiplication)is,
(5i + 3j) (3i - 5j) = 5(3) + 3(-5) = 15 -15 = 0
Try the wikipedia and mathworld articles. Hope this helps!

Answer 3

You can tell just by looking.

5i+3j and 3i-5j

Notice the 3 and 5 switched places and the sign changed. They are orthogonal.

Picture this on a graph.
Start at (0,0) and go 5 right and 3 up----5i+3j. draw a line from (0,0) to (5,3).

Next go from (0,0) to 3 right and 5 down----3i-5j. Draw your second line from (0,0) to (3,-5). You will see that these line are perpendicular.

PROOF

The 5i+3j vector forms a certain angle with respect to the x-axis. The 3i-5j vector forms the same angle with respect to the y-axis. You can see that these vectors will each form congruent right triangles, each with legs of 3 and 5, so we know the corresponding angle are equal.
Since the x- and y-axes are perpendicular, the vectors which are equally offset form their respective axes must be perpendicular.

Answer 4

a) and c)
a) 5i+3j dot c) 3i-5j =15 -15 =0

Answer 5

2 vectos are ortogonal when aa' + bb' = 0

So, if you consider the 2 first, a = 5, b = 3, a' = 1, b'= 1, you notice that 5.1 + 3.1 isnt 0

So, go on and find the right answer

Ana

Answer 6

vector a) and vector c) are perpendicular to each other.
(a1,b1) , (a2,b2) are perpendicular if
a1. a2+b1.b2=0
cos 90 =0

Answer 7

All of them

i dont know how but i am just guessing