Using Inner Product Properties to Solve Vector Problems

The remaining one is just plugging in the numbers.In summary, the problem involves finding the magnitude of a unit vector in terms of the given dot products between a, b, and c. This can be done using the properties of the dot product, specifically its linearity in both arguments for real scalars. By plugging in the given values, the magnitude of a can be evaluated as √(14)/8.
  • #1
Blackbear38
1
0
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.]
 
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  • #2
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.
 
  • #3
Blackbear38 said:
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.
 
  • #4
Sorry for this question but are you sure you know what a unit vector is?
 
  • #5
You need to rely more on the basic properties of the inner product. The inner product is linear in both its arguments for real scalars. That should give you almost all the answers.
 

1. What is the dot product of two vectors?

The dot product of two vectors is a mathematical operation that results in a scalar quantity. It is calculated by multiplying the corresponding components of the two vectors and adding them together.

2. How do you find the dot product of two vectors?

To find the dot product of two vectors, you need to first ensure that the two vectors have the same number of dimensions. Then, multiply the corresponding components of the two vectors and add them together. The resulting value is the dot product of the two vectors.

3. What is the purpose of the dot product in vector problems?

The dot product is used in vector problems to determine the angle between two vectors, the projection of one vector onto another, and the length of a vector. It also has applications in physics, engineering, and computer graphics.

4. Can the dot product of two vectors be negative?

Yes, the dot product of two vectors can be negative. This occurs when the angle between the two vectors is greater than 90 degrees. In this case, the dot product represents the negative of the magnitude of the projection of one vector onto the other.

5. What is the difference between dot product and cross product?

The dot product and cross product are two different mathematical operations involving vectors. The dot product results in a scalar quantity, while the cross product results in a vector quantity. Additionally, the dot product is commutative (order of vectors does not matter), while the cross product is anti-commutative (order of vectors matters).

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