Dot product vectors problem

Blackbear38
Summary:: I need to solve a problem for an assignment but just couldn't find the right approach. I fail to eliminate b or c to get only the magnitude of a.

Let a, b and c be unit vectors such that a⋅b=1/4, b⋅c=1/7 and a⋅c=1/8. Evaluate (write in the exact form):
- ||4a||
- 3a.5b
- a.(b-c)
- (a+b+c).(a-b)

What I first did was ab.ac = a^(2).bc then substitute values of ab, ac, and bc, but I cannot confirm that this is the correct approach. Hence, I found:

ab.ac = a^(2).bc
(1/4)(1/8)=a^(2)(1/7)
a^(2) = 7/32
Hence, ||a|| = sqrt(14)/8

I really hope that this doubt can be clarified for all the parts of my question. Thanks!

[Moderator's note: moved from a technical forum.]

What you need to know is

a is a unit vector, so what is its magnitude?
scalars and vectors are commutative.
The scalar product is commutative and distributive.

The rest is given.

Mentor
Summary:: I need to solve a problem for an assignment but just couldn't find the right approach. I fail to eliminate b or c to get only the magnitude of a.

Let a, b and c be unit vectors such that a⋅b=1/4, b⋅c=1/7 and a⋅c=1/8. Evaluate (write in the exact form):
- ||4a||
- 3a.5b
- a.(b-c)
- (a+b+c).(a-b)

What I first did was ab.ac = a^(2).bc
I have no idea why you did this, and if '.' means "dot product" the above makes no sense.
##a \cdot b## is a number (given) and ##a \cdot c## is also a number (also given). The dot product is defined for vectors, but not plain old numbers.
Blackbear38 said:
Then substitute values of ab, ac, and bc, but I cannot confirm that this is the correct approach.
It's not.
Blackbear38 said:
Hence, I found:

ab.ac = a^(2).bc
(1/4)(1/8)=a^(2)(1/7)
a^(2) = 7/32
Hence, ||a|| = sqrt(14)/8

I really hope that this doubt can be clarified for all the parts of my question. Thanks!

[Moderator's note: moved from a technical forum.]
Following up on @gleem's comments, you need to be looking at the properties of the dot product, such as ##ku \cdot v = k u \cdot v## and ##u \cdot (v +w) = u \cdot v + u \cdot w##, etc. This is a very easy set of problems if you know these properties, plus the fact that a, b, and c are all unit vectors.

Homework Helper
Gold Member
Sorry for this question but are you sure you know what a unit vector is?