# Dot Product Proof: Prove |a|^2 = 2|b|^2

• EmilyHopkins
In summary, the problem is asking to express the vectors PQ and PR in terms of a and b, given that point P and Q have position vectors a + b and 3a - 2b respectively, and that OPQR is a parallelogram. By using two scalar products, it is shown that if OPQR is a square, then |a |2 = 2 |b |2. The solution involves taking the dot product of OP and PQ to obtain 2a2 -ab -3b2 = 0, which can be rearranged to ab = 2a2 - 3b2. Similarly, the dot product of PR and OQ is used to get 3a
EmilyHopkins

## Homework Statement

The point P and Q have postion vectors a + b, and 3a - 2b respectively, relative to the origin O.Given that OPQR is a parallelogram express the vector PQ and PR in terms of a and b. By evaluating two scalar products show that if OPQR is a square then |a |2 = 2 |b |2

## The Attempt at a Solution

OP = a + b

OQ = 3a -2b

PQ = OR= PO + OQ = -OP +OQ = -(a + b) + (3a - 2b) = 2a - 3b

PR = PO + OR = -OP + OR = -(a + b) + (2a - 3b) = a - 4b

So now the question says use two scalar products to show |a |2 = 2 |b |2. I'm assuming since the question ask for these two vectors in terms of a, and b that you will have to utilize it to get the result. So I drew out the square for a visualization.

So since its a square it means the dot product of

OP.PQ= 0
(a + b)(2a - 3b) = 2a2 -ab -3b2 = 0

ab = 2a2 -3b2

PR.OQ= 0 (Since their perpendicular)
(a-4b)(3a - 2b) =0
3a2 - 14ab +8b2=0
3a2 -14(2a2 -3b2) +8b2=0
3a2 -28b2 +42a2 + 8b2=0
-25a2+50b2=0
25a2=50b2
a2= 2b2

Is their a faster way to work it or is this correct ?
?

What is the relationship between a and b? Are they just two arbitrary, non parallel, vectors? Orthogonal unit vectors? What?

Does the diagram come with the question or is it one you have drawn assuming that the parallelogram is a square? Note: the angles in a parallelogram do not have to be 90 degrees. Q does not have to be on the opposite corner to O. (Off the question alone, I'd have taken OP and OQ to be adjacent sides, and OR=OP+OQ.)

Simon Bridge said:
What is the relationship between a and b? Are they just two arbitrary, non parallel, vectors? Orthogonal unit vectors? What?

Does the diagram come with the question or is it one you have drawn assuming that the parallelogram is a square? Note: the angles in a parallelogram do not have to be 90 degrees. Q does not have to be on the opposite corner to O. (Off the question alone, I'd have taken OP and OQ to be adjacent sides, and OR=OP+OQ.)

I'm assuming a and b are two non-parallel arbitrary vectors, as the question never specified what their relationship was. The diagram was used to visualize and help solve the second part of the question where they stated that if the parallelogram OPQR was a square, show by using two scalar products that |a |2 = 2 |b |2 .

Oh I missed the "if it were square" part.
The relationship means that a is the hypotenuse of a 1-1-root-2 triangle.

If you swap the positions of Q and R on your square, will it still fit the description?

In the following:
(a + b)(2a - 3b) = 2a2 -ab -3b2 = 0
you only expanded to three terms;
since a and b are vectors, a.a = a2 is a little ambiguous;
I think you need to choose a notation that distinguishes between the length of a vector and the vector itself. In the above case:
(a+b).(2a-3b)=2a.a+2b.a-3a.b-3b.b=2|a|2-3|b|2+... how would you handle the mixed dot products? Is a.b the same as b.a?

Other than that - I think you have the actual method intended.
The only wrinkle remaining is the thing about the position of Q and R.
Does it make a difference?

## 1. How is the dot product defined?

The dot product is a mathematical operation that takes two vectors and returns a scalar value. It is calculated by multiplying the corresponding components of the two vectors and then summing the products.

## 2. What is the geometric interpretation of the dot product?

The dot product can be interpreted as the product of the magnitudes of two vectors and the cosine of the angle between them. This means that the dot product is largest when the two vectors are parallel, and smallest when they are perpendicular.

## 3. How does the dot product relate to vector length?

The dot product is related to vector length through the Pythagorean theorem. Specifically, the dot product of a vector with itself (also known as the squared length or norm) is equal to the sum of the squares of its components.

## 4. How can the dot product proof be used to show that |a|^2 = 2|b|^2?

The dot product proof shows that the squared length of vector a (|a|^2) is equal to twice the squared length of vector b (2|b|^2) by expanding the dot product of a with itself and then using the properties of vector length and the Pythagorean theorem to simplify the equation.

## 5. What implications does the dot product proof have in vector algebra?

The dot product proof is a useful tool in vector algebra as it allows us to relate the lengths of two vectors through their dot product. It also demonstrates the geometric relationship between the dot product and vector length, which can be applied in various mathematical and scientific fields.

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