Angle between 2 Vectors in 3D:

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Find the angle between the vectors Equation of vector

, and

How do we proceed?

We can set up the dot product of the vectors in two ways and solve for the cosine of the angle.

State those two ways in terms of the components and the lengths

Dot product in components and lengths and angle

Solve for the cosine.

Expression for the cosine of the angle between two vectors

Can we express

in terms of their components, too?

Yes.

State those expressions.

and

How do we eventually obtain the angle?

We use the inverse cosine to find the angle.

Let's plot this on paper, using the usual 3-D axes with the x-axis out of the plane of the diagram. Draw the axes and then click "Next" to check your work.

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Now show the vector A, based at the origin. Then check your work by clicking "Next".

Note the use of color for the coordinates of the head of the vector. alt="Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!

Now show the vector B, based at the origin. Then check your work by clicking "Next".

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Let's discuss the calculations summarized above in more detail. How do we find the Dot Product?

We find the Dot Product from the sum of the products of the components.

Set that up in terms of the general components

of the vectors.