# Finding the Angle Between Vectors a and b

• Jane K
In summary, the conversation discusses finding the angle between two perpendicular vectors, a and b, with lengths of 2 and 1 respectively. The dot product method is used to find the angle, with the final equation being arccos[(a.b)/(|a||b|)].
Jane K
1. Homework Statement
The vector a=2 and vector b=1. The vectors a+5b and 2a-3b are perpendicular. Determine the angle between a and b .

## Homework Equations

The dot product a•b=lallblcosθ

## The Attempt at a Solution

I've tried a few things but none of it really makes sense. I'm worried that maybe this question doesn't call for the dot product method but I've become fixed on it.

Jane K said:
1. Homework Statement
The vector a=2 and vector b=1. The vectors a+5b and 2a-3b are perpendicular. Determine the angle between a and b .

## Homework Equations

The dot product a•b=lallblcosθ

## The Attempt at a Solution

I've tried a few things but none of it really makes sense. I'm worried that maybe this question doesn't call for the dot product method but I've become fixed on it.

You don't have a vector a = 2 and b = 1, those are scalars. Perhaps you mean their magnitudes are 2 and 1? What do you get if you dot a+5b and 2a-3b together?

"the vectors a and b have lengths 2 and 1, respectively." I'm trying to find the angle between the two. The dot product for the perpendicular vectors a+5b and 2a-3b would be zero... I am stuck:P.

Jane K said:
"the vectors a and b have lengths 2 and 1, respectively." I'm trying to find the angle between the two. The dot product for the perpendicular vectors a+5b and 2a-3b would be zero... I am stuck:P.

Show us what you get in terms of a and b when you dot those two vectors together and set it equal to 0.

Jane K said:
"the vectors a and b have lengths 2 and 1, respectively." I'm trying to find the angle between the two. The dot product for the perpendicular vectors a+5b and 2a-3b would be zero... I'm stuck:P.
"FOIL" works for vectors .

$\left(\vec{a}+\vec{b}\right)\cdot\left(\vec{c}+ \vec{d}\right)=\vec{a}\cdot\vec{c}+\vec{a}\cdot \vec{d}+\vec{b}\cdot\vec{c}+\vec{b}\cdot\vec{d}$​

what SammyS means is that the dot product is bilinear, it is linear in each variable:

if a,b,c are vectors, and r is a scalar:

a.(b+c) = a.b + a.c
(a+b).c = a.c + b.c

a.(rb) = r(a.b)
(ra).b = r(a.b)

also, a.b = b.a (the dot product is symmetric).

thus (a+5b).(2a-3b) = 2(a.a) + 5(b.a) - 3(a.b) - 15(b.b)

a.a = |a|2, for any vector a.

you are given |a| and |b|, and you are given that the dot product (a+5b).(2a-3b) = 0.

if you can deduce what a.b is, then you can figure out the angle:

θ = arccos[(a.b)/(|a||b|)]

Thank-you everyone!
This really helped:)

## 1. What is the formula for finding the angle between two vectors a and b?

The formula for finding the angle between two vectors a and b is given by:
cosθ = (a • b) / (|a| * |b|), where θ is the angle between the two vectors and (a • b) represents the dot product of the two vectors.

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

To find the dot product of two vectors, you multiply the corresponding components of the two vectors and then add them together. For example, if vector a = (a1, a2, a3) and vector b = (b1, b2, b3), then the dot product of a and b is given by:
a • b = a1*b1 + a2*b2 + a3*b3.

## 3. Can the angle between two vectors ever be negative?

No, the angle between two vectors is always positive. This is because the range of the inverse cosine function (used to find the angle) is between 0 and π, which means the resulting angle will always be positive.

## 4. What is the significance of finding the angle between two vectors?

Finding the angle between two vectors is important in understanding the relationship between the two vectors. It can also be used to determine if the vectors are perpendicular (angle = 90°), parallel (angle = 0° or 180°), or at an acute or obtuse angle to each other.

## 5. Is there a special way to find the angle between two vectors in 3-dimensional space?

Yes, the formula for finding the angle between two vectors in 3-dimensional space is the same as the formula for 2-dimensional space, except now the dot product is replaced by the cross product. The formula is:
cosθ = (a × b) / (|a| * |b|), where θ is the angle between the two vectors and (a × b) represents the cross product of the two vectors.

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