Find Angles of Vector A with Coordinate Axes

But it doesn't go far enough.The author of the solution is using the fact that i•i = 1, j•i = k•i = 0 to do the calculation.For the first part, (3i - 6j + 2k)•i = 3i•i - 6j•i + 2k•i, he is using the distributive property to multiply each term in the parentheses by i. The result is 3i•i - 6j•i + 2k•i.In the second part, he uses the definitions of i, j, k to do the calculations. i•i = 1 (since i•i
  • #1
MaxManus
277
1

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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  • #2
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:

[tex]\vec{A}\cdot \hat i = A \cos\theta[/tex]

What do these equal?
i • i = ?
j • i = ?
k • i = ?
 
Last edited:
  • #3
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 ?
 
  • #4
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.
 
  • #5
Thanks
 
  • #6
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 [itex]\cdot[/itex] 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.
 
  • #7
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.)
 

Related to Find Angles of Vector A with Coordinate Axes

What is the definition of "Find Angles of Vector A with Coordinate Axes"?

"Find Angles of Vector A with Coordinate Axes" refers to the process of determining the angles between a vector and the X, Y, and Z axes in a three-dimensional coordinate system. This can be done using trigonometric functions and the coordinates of the vector.

Why is it important to find the angles of a vector with coordinate axes?

Knowing the angles of a vector with coordinate axes is important in many fields of science, including physics and engineering. It allows us to understand the direction and orientation of the vector in relation to the coordinate system, which is crucial in solving many problems and making accurate measurements.

What are the steps to find the angles of a vector with coordinate axes?

The first step is to determine the coordinates of the vector. Then, using the coordinates, calculate the length of the vector and the angles it makes with each of the axes using trigonometric functions. The final step is to interpret the results in terms of direction and orientation.

Can the angles of a vector with coordinate axes be negative?

Yes, the angles of a vector with coordinate axes can be negative. This happens when the vector points in the opposite direction of the positive axis. For example, if the vector points in the negative X direction, the angle with the X axis would be -180 degrees or -π radians.

Are there any applications of finding the angles of a vector with coordinate axes?

Yes, there are many applications of finding the angles of a vector with coordinate axes. For example, in physics, it is used to calculate the trajectory of objects in motion, while in engineering, it is used to determine the direction and magnitude of forces acting on structures. It is also used in computer graphics to determine the orientation of 3D objects on a screen.

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