2 vectors are perpendicular if?

Question

Two vectors <V1, V2> and <W1, W2> are perpendicular if

 

a. their cross product equals zero
b. their inner product equals zero
c. their inner product is v1w1 +v2w2
d. their dot product is equal to their cross product

Answer 1

b. Their inner product is zero

 

Take as example the vectors (1,0,0) and (0,1,0)...definitely perpendicular. The dot product is the most standard example of an inner product. So, we dot: 1*0 + 0*1 + 0*0 = 0. If I forget a property like this, I often use example to try and figure it out.

 

You can also use these example vectors to check that A and D are not true. A is not true since the cross product of these vectors is (0,0,1), not zero. Part D doesn't even make sense, since dot product is a scalar while cross product is a vector.

 

Part C is funny. The expression v1w1+v2w2 is the definition of the dot product of the vectors <v1, v2> and <w1, w2>. The dot product is an example of an inner product. Saying the inner product equals the dot product just lets us know how to inner product two vectors...it says nothing about determining if two vectors are perpendicular.

 

Hope this helps you sort through what these answer choices mean. Let me know if you have any further questions, or if you want to see a proof of why part b is true. I am available for online tutoring!

Answer 2

For a problem like this, you need to start with the definition.

 

What is a cross product? If we look at the definition, we will see that "The cross product a × b is defined as a vector c that is perpendicular to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span." That means that if we have a vector (1,0,0) (the vector on x-axes) and the vector (0,1,0) (y-axes), the cross product would be the vector (0,0,1) (z-axes), not zero.

 

Let's find a definition for an inner product. That is a number. And that is v1w1 +v2w2. So, let's take 2 perpendicular vectors (1,0) and (0,1). What is the inner product? 1*0+0*1=0. This is probably our answer.

 

c is just a definition of inner product in general, so it is not our answer

 

d is not what we need because the number is never equal to the vector.