Vector 'B' When Vector Product with 'A' is Zero

[draotic]

Homework Statement

Vector 'A' is along positive z-axis and its vector product with another vector 'B' is zero , then vector 'B' could be ..
a) i + j
b) 4i
c) i + k
d) -7k
.

Homework Equations

The Attempt at a Solution

 

[wbandersonjr]

What do you think? What have you tried, or what equations are useful?

 

[tiny-tim]

hi draotic!

what is the cross product (vector product) of A with each of those four vectors? ​

 

[draotic]

What do you think? What have you tried, or what equations are useful?

i m trying 2 use distributive law
"A = i - j
B = i + k
AxB = (i - j) x (i + k) = ixi + ixk - jxi - jxk = (0 + (-j ) - (-k) - i)
= -i - j + k"
but can't get the anser

 

[draotic]

hi draotic!

what is the cross product (vector product) of A with each of those four vectors? ​

yup..that might work
but i don't think in an entrance xam ,i will hve enought time to find the vector product of each option with A...
is there something direct?
thanks

 

[draotic]

ok i think i got the solution...please check if its right
AxB=0 (given)
thereferore ABsintheta=0
hence theta=0 degree
this shows that A and B are collinear i.e,they must lie along same axis
...
so the only option with same axis as A is -7K
so its (d)
right?

 

[wbandersonjr]

Yes there is something more direct. When a cross product comes out to zero, it tells you something special about the relationship between the two vectors. What does it mean is the cross product is zero? Hint: think about another way to write the cross product of two vectors, and think about the angle.

 

[wbandersonjr]

Haha, I didnt respond fast enough. That is exactly right, that is what is important to see about this problem!

 

[wbandersonjr]

hence theta=0 degree

This is not necessarily true. $\Theta$ could be 180 also, or someother multiple of $\pi$.

But since sin$\Theta$ equals zero, we know that the two vectors are parallel or antiparallel.

 

[draotic]

This is not necessarily true. $\Theta$ could be 180 also, or someother multiple of $\pi$.

But since sin$\Theta$ equals zero, we know that the two vectors are parallel or antiparallel.

can u think of anything whiich gets me direct anser of -7k??

 

[wbandersonjr]

Your way of answering is the direct way. This question is not so much about you computing a cross product as it is about you understanding that a cross product equal to zero means that the vectors are parallel. This is the concept that the question is reinforcing. You could do this by brute force, but that is time consuming and unnecessary if you understand the cross product well.

 

[draotic]

Your way of answering is the direct way. This question is not so much about you computing a cross product as it is about you understanding that a cross product equal to zero means that the vectors are parallel. This is the concept that the question is reinforcing. You could do this by brute force, but that is time consuming and unnecessary if you understand the cross product well.

thanx for the help sir

 

[wbandersonjr]

Anytime.

 

[DHeshMan]

Ok, to put it very simply, since vector A is along the z axis, its definition will be some magnitude in the z direction, with no magnitude in x or y. A = mk. Taking the cross product of this vector with any vector in the same direction will equal zero. Or alternatively, taking a cross product with any vector that is not in the same direction will result in a non-zero answer. So seeing that a) b) and c) all have some component of i or j, a cross product with any of them will result in a non-zero answer. A cross product with any vector that is a scalar multiple of the original vector (same direction) is zero, so if vector A was a little more complicated than just being along the z-axis, the problem would still be quite simple to quickly deduce the answer.

If you were a gambling person, and didn't know if they were referring to the cross product or the dot product in this question, or if you didn't know that the cross product is zero when parallel or perpendicular, you can deduce that by seeing more than one answer was perpendicular, one was neither perpendicular nor parallel, and only one was parallel, you can infer that d) would be your correct answer.