Algebraic Dot Products! Find a dot b if a=i-2j+3k, b=2j-3k

[iksaxophone]

Hey guys, I've been trying to figure out how to calculate dot products using variables and I'm pretty stuck. I get how to simplify using the special properties when given points in space i.e. a=(1,4,6), b=(3,7,12). However, there are also problems such as:

Find a dot b if

a=i-2j+3k

b=2j-3k

My textbook has no purely algebraic examples even though it gives purely algebraic questions. I tried to use the distributive property and wound up with:

a dot b=(i-2j+3k)(2j-3k)
=(i dot 2j)+(i dot -3k)+(-2j dot 2j)+(-2j dot -3k)+(3k dot 2j)+(3k dot -3k)

This appears to just leave me with more dot products to calculate and doesn't simplify anything. Any suggestions would be much appreciated, and thank you in advance!

 

[tkhunny]

Hey guys, I've been trying to figure out how to calculate dot products using variables and I'm pretty stuck. I get how to simplify using the special properties when given points in space i.e. a=(1,4,6), b=(3,7,12). However, there are also problems such as:

Find a dot b if

a=i-2j+3k

b=2j-3k

My textbook has no purely algebraic examples even though it gives purely algebraic questions. I tried to use the distributive property and wound up with:

a dot b=(i-2j+3k)(2j-3k)
=(i dot 2j)+(i dot -3k)+(-2j dot 2j)+(-2j dot -3k)+(3k dot 2j)+(3k dot -3k)

This appears to just leave me with more dot products to calculate and doesn't simplify anything. Any suggestions would be much appreciated, and thank you in advance!

Why does your Dot Product result in a Vector? You appear to be treating your 3-vectors like trinomials.

"Purely Algebraic" doesn't mean anything. Please reread the definition of a Dot Product and give it another go.

 

[Dr.Peterson]

Hey guys, I've been trying to figure out how to calculate dot products using variables and I'm pretty stuck. I get how to simplify using the special properties when given points in space i.e. a=(1,4,6), b=(3,7,12). However, there are also problems such as:

Find a dot b if

a=i-2j+3k

b=2j-3k

My textbook has no purely algebraic examples even though it gives purely algebraic questions. I tried to use the distributive property and wound up with:

a dot b=(i-2j+3k)(2j-3k)
=(i dot 2j)+(i dot -3k)+(-2j dot 2j)+(-2j dot -3k)+(3k dot 2j)+(3k dot -3k)

This appears to just leave me with more dot products to calculate and doesn't simplify anything. Any suggestions would be much appreciated, and thank you in advance!

It will help if you show us how your textbook defines the dot product, including any alternative ways they show to evaluate it.

What you did would be a valid start (the long way), if you know what i.i, i.j, and so on are. These are all scalars, and many terms would drop out, yielding a very simple scalar answer.

But there are many ways you could have been introduced to the concept, and some lead more directly to the answer than others. In order to help, we need to see what you know about it.

Are you aware that, for example, i is the same as (1,0,0), and j is a name for (0,1,0)? Do you know that (1,4,6) is the same as i + 4j + 6k?

 

[HallsofIvy]

I think that what iksaxaphone is doing is saying that (ai+ bj+ ck) dot (xi+ yj+ zk)= ai dot (xi+ yj+ zk)+ bj dot (xi+ yj+ zk)+ ck dot (xi+ yj+ zk)= ai dot xi+ ai dot yj+ ai dot zk+ bj dot xi+ bj dot yj+ bj dot zk+ ck dot xi+ ck dot yj+ ck dot zk= ax (i dot i)+ay (i dot j)+ az(i dot k)+ bx (j dot i)+ by (j dot j)+ bz (j dot k)+ cx (i dot i)+ cy (k dot j)+ ck (k dot k)

That is all true but immediately simplifies to ax+ by+ ck because
i dot i= j dot j= k dot k= 1 and
i dot j= i dot k= j dot i= j dot k= k dot i= k dot j= 0.