Find the angle between two vectors

1. Sep 22, 2011

jamesbrewer

1. The problem statement, all variables and given/known data

Two vectors A and B have the same magnitude of 5.25. If the sum of these two vectors gives a third vector equal to 6.73j, determine the angle between A and B.

2. Relevant equations

For some vector $\vec{R}$: $|\vec{R}| = \sqrt{R_x^2 + R_y^2}$

3. The attempt at a solution

I feel like I need to find the components of A and B, but I can't figure out how to do that with only the information given -- is it even possible?

I think the magnitude of the third vector is 6.73. Since there was no $\hat{i}$ term given for $\vec{R}$, I would say that $|\vec{R}| = \sqrt{R_x^2 + R_Y^2} = \sqrt{6.73^2} = 6.73$.

Given the components of A and B, this would be easy to solve. Without them I am utterly lost. What am I missing?

2. Sep 22, 2011

PeterO

The vectors A and B, along with the resultant will form a triangle - probably not right angled - so you can use the cosine Rule to solve.

a2 = b2 + c2 - 2.b.c.cos(A)

3. Sep 22, 2011

jamesbrewer

What are a, b, and c? The lengths of the sides of the triangle? If so, would that mean that a = |A|, b = |B|, and c = |C| (the resultant)?

4. Sep 22, 2011

PeterO

a, b & c are the sides,
A is the angle opposite side a

[so a, b & c could represent vectors A, B and C but not necessarily, depends which angle you are looking for]

Last edited: Sep 22, 2011
5. Sep 22, 2011

jamesbrewer

How can I form a triangle if I don't know anything other than the vector's magnitude? I have nothing to tell me what direction it points in.

6. Sep 22, 2011

PeterO

You can stand a triangle up any way you like - its sides are still the same length and its angles are still the same size. A name like "the base" might apply to a different side, depending which way you arrange it, but that should not be a problem .. what's in a name?

7. Sep 22, 2011

jamesbrewer

Here's what I've got:

$|\vec{C}|^2 = |\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}|cos\theta$

$6.73^2 = 5.25^2 + 5.25^2 - 2(5.25)^2 cos \theta$

$45.29 = 27.56 + 27.56 - 2(27.56) cos \theta$

$45.29 = 55.12 - 55.12 cos \theta$

$45.29 - 55.12 = - 55.12 cos \theta$

$-9.83 = -55.12 cos \theta$

$\frac{-9.83}{-55.12} = cos \theta$

$\theta = cos^-1 0.178$

$\theta = 79.75^o$

My answer wasn't correct though, where did I go wrong?

Last edited: Sep 22, 2011
8. Sep 22, 2011

Staff: Mentor

One approach which you might want to consider is to look at the relationships between the vector components in terms of simultaneous equations. Suppose that the two initial vectors are A and B and the resultant is C.

Since C contains no x-component and has only a y-component, the sum Ax + Bx must be zero, or in other words, Bx = -Ax. Similarly, Ay + By must be Cy. You also have |A| = |B| = 5.25, so that's another pair of relationships. You have four unknowns (really three when you consider that the x-components of A and B are equal and opposite) and plenty of interrelationships to use to solve for them.

9. Sep 22, 2011

PeterO

How incorrect was your answer? You have rounded off all the way through which could make a small difference

10. Sep 23, 2011

jamesbrewer

Unfortunately I have no idea. All I was told was "Incorrect answer."

11. Sep 23, 2011

Staff: Mentor

Let's step back from the problem and see if we can make additional simplifying deductions before invoking formulas.

Since the resultant of vectors A and B is a vector with only a y-component (6.73j), then A and B must have equal and opposite x-components. Further, since A and B have equal magnitudes (5.25), this then forces their y-components to be equal also. Why not equal and opposite you say? Because then their sum would be zero rather that +6.73.

So A and B "straddle" the positive y-axis, and make equal angles with that axis. The sum of their y-components is 6.73. Now it's time to write formulas. If you let $\theta$ be the angle between either A or B and the Y-axis, what is an expression for the y-component of A or B? If double that is 6.73, can you solve for $\theta$? What then is the angle between A and B?

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12. Sep 23, 2011

PeterO

Draw a diagram of the vectors, like gneil's and you will see that they actually want (180 - 79) degrees - with the decimal bits.
Rounding off needs to be addressed too.
gneil's method is excellent also - even preferable!

13. May 15, 2013

Karthik k

Three vectors sum to zero. The magnitude of 2 vectors is equal and the third one is root 2 times the magnitude of the equal vectors. Find the angle between the three vectors. Plzz help me on dis one..:)

14. May 15, 2013

PeterO

Since the three vectors will make up a triangle - so you end up where you started with a vector sum of zero - you should recognise the magnitudes 1,1,√2 as the sides of a very common triangle in trigonometry, from which we derive the exact vale of sin, cos and tan of a particular angle.
The other triangle used in trigonometry is known as the 2,1,√3 triangle [not that it relates directly to this question].