Website Title: Calculating Angles Between Vectors in Three Dimensions

In summary, the correct method for calculating the angles between a vector and the positive x-, y-, and z- directions is to take the cosines of the components in each direction. This is because cosines represent the ratios of the components to the total magnitude, while sines represent the ratios of the components to the magnitude in the perpendicular direction.
  • #1
enryiggins
2
0
I am trying to teach myself AS maths. This is from Introducing mechanics a txt book by Jefferson and Beadsworth. exercise 2C question 1 part vi)

Question given vector b=6i-3j-2k;
|b| is established as = 7 by calculation

Question
Calculate the angles between b and positive x-, y- and z- directions.

My answer: these are
sine alpha =6/7 = 59 degrees - subtract from 180 to give angle from z axis


Sine beta =2/7 = 16.6 degrees - this is angle from x axis

Sine theta = 3/7 = 25.4 degrees. Need to add 90 degrees to give angle from y axis.

However these are not the correct answers. The correct answers appear to be cosines of the above. This does not make sense to me can you expalin?
 
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  • #2
Once you normalize the vector to unit length, the coeficients of (i,j,k) are the cosines with respect to the axes (6/7,-3/7,-2/7).
 
  • #3
Welcome to PF!

oh-oh-oh-enryiggins! Welcome to PF! :smile:
enryiggins said:
I am trying to teach myself AS maths.

Just you wait, enryiggins, just you wait.
You'll be sorry, but your tears :cry: will be too late. :wink:​
… Calculate the angles between b and positive x-, y- and z- directions.

My answer: these are
sine alpha =6/7 = 59 degrees - subtract from 180 to give angle from z axis


Sine beta =2/7 = 16.6 degrees - this is angle from x axis

Sine theta = 3/7 = 25.4 degrees. Need to add 90 degrees to give angle from y axis.

However these are not the correct answers. The correct answers appear to be cosines of the above. This does not make sense to me can you expalin?

The component in a direction is always multiplied by the cos, not the sin.

After all, cos0º = 1, and if the angle is 0º, then the whole of the vector is the component in that direction … so it has to be cos, doesn't it? :wink:
 
  • #4
Many thanks for your help. My drawing was not correct hence the confusion.
 

FAQ: Website Title: Calculating Angles Between Vectors in Three Dimensions

What are vectors in three dimensions?

Vectors in three dimensions are mathematical objects that have both magnitude and direction. They are commonly represented as arrows in three-dimensional space, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction.

How are vectors in three dimensions different from vectors in two dimensions?

Vectors in three dimensions are different from vectors in two dimensions because they have three components (x, y, and z) instead of just two (x and y). This means that they can represent movement in three-dimensional space, while vectors in two dimensions can only represent movement in a plane.

How do you represent a vector in three dimensions?

A vector in three dimensions can be represented in multiple ways, including using Cartesian coordinates (x, y, z), unit vectors (i, j, k), or a combination of both. The vector can also be represented as a magnitude and direction, or as a linear combination of basis vectors.

What is the difference between a position vector and a direction vector?

A position vector represents a specific location or point in space, while a direction vector represents a direction or displacement from one point to another. Position vectors have a fixed starting point (the origin), while direction vectors do not have a fixed starting point.

How do you perform vector operations in three dimensions?

Vector operations in three dimensions, such as addition, subtraction, and scalar multiplication, can be performed by adding or subtracting the corresponding components of the vectors. Vector dot and cross products can also be used to perform operations between vectors in three dimensions.

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