Find Angles of Vector A with Coordinate Axes

AI Thread Summary
To find the angles that vector A = 3i - 6j + 2k makes with the coordinate axes, the dot product is used. The calculation shows that A • i = 3, derived from the expression (3i - 6j + 2k) • i, where i • i = 1 and other terms equal zero. The magnitude of vector A is calculated as 7, leading to the relationship cos(a) = 3/7. This allows for solving for the angle a using the cosine function. Understanding the dot product and its relationship to vector projections is crucial for this problem.
MaxManus
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Homework Statement


Find the angles which the vector A = 3i -6j +2k makes with the coordinate axes




The Attempt at a Solution


Let a, b, c be the angles which A makes with the positive x,y,z axes.
A• i = (A)(i)cos(a) = 7*cos(a)


The Solution says:
Ai = (3i - 6j + 2k)• i = 3i• i -6j• i + 2k•i = 3

And I do not understand how they get 3 as the answer.
 
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MaxManus said:

The Attempt at a Solution


Let a, b, c be the angles which A makes with the positive x,y,z axes.
A• i = (A)(i)cos(a) = 7*cos(a)
You need to take the product of the vector A with i, not the magnitude of A:

\vec{A}\cdot \hat i = A \cos\theta

What do these equal?
i • i = ?
j • i = ?
k • i = ?
 
Last edited:
I was supposed to write A• i = (A)(1)cos(a) = sqrt(3**2 + (-6)**2 + 2**2) = 7*cos(a)

second question:

had to check the book and it says
i• i = 1
j• i = j• k = 0

ah, thanks

So
A• i = (3i - 6j + 2k)• i = 3i• i -6j• i + 2k•i = 3 - 0 + 0 = 3 ?
 
MaxManus said:
I was supposed to write A• i = (A)(1)cos(a) = sqrt(3**2 + (-6)**2 + 2**2) = 7*cos(a)
OK.

So
A• i = (3i - 6j + 2k)• i = 3i• i -6j• i + 2k•i = 3 - 0 + 0 = 3 ?
Exactly. Now use this result to solve for the angle a in your first equation.
 
Thanks
 
MaxManus said:

Homework Statement


Find the angles which the vector A = 3i -6j +2k makes with the coordinate axes




The Attempt at a Solution


Let a, b, c be the angles which A makes with the positive x,y,z axes.
A• i = (A)(i)cos(a) = 7*cos(a)
There's nothing wrong with the above, as far as it goes. In addition to the coordinate definition of the dot product, there is the definition that involves the magnitudes of the vecctors and the angle between them.

In this case cos(a) = (length of the projection of A onto the x-axis)/(magnitude of A) = 3/7.

So A \cdot i = 7 * cos(a) = 7 * 3/7 = 3
MaxManus said:
The Solution says:
Ai = (3i - 6j + 2k)• i = 3i• i -6j• i + 2k•i = 3

And I do not understand how they get 3 as the answer.
 
Mark44 said:
There's nothing wrong with the above, as far as it goes.
That's true. (I could have explained things better in my first response.)
 
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