Find Angle of Vector h: 3ax+5ay-8az

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In summary, to find the angle between a vector h and the x, y, and z axes, you can take the dot product of h with each respective unit vector. Alternatively, you can find the magnitude of h and divide it by the magnitude of the vector formed by the x, y, and z components of h to determine the cosine of the angle in each direction.
  • #1
zekester
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i have a vector h whose value is 3ax+5ay-8az where a is the unit vector in the direction beside it, how would i find the angle between this vector and the x,y,and z axes
 
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  • #2
where a is the unit vector in the direction beside it,

Do you mean h = 3x + 5y - 8z ?

In any case, in order to find the angle between vectors, take the dot product.

Eg. Angle between h and x- axis would be given by:

h . (unit vector x) = (magnitued of h) * cos (theta)
 
  • #3
Or, you can find the sq. root of the sum of the squares (3^2+5^2+(-8)^2)^0.5

and then for each direction divide the magnitude by this value and it will give you the cos for each angle
 

What is the formula for finding the angle of a vector?

The formula for finding the angle of a vector is θ = cos^-1 (v1⋅v2 / ||v1|| ||v2||), where v1 and v2 are the two vectors and || || represents the magnitude of the vector.

How do you find the angle of a vector in 3-dimensional space?

To find the angle of a vector in 3-dimensional space, first calculate the dot product of the two vectors. Then, divide the dot product by the product of the magnitudes of the two vectors. Finally, take the inverse cosine of this value to find the angle.

What is the significance of the direction cosines in finding the angle of a vector?

The direction cosines, represented by a, b, and c in the formula, indicate the direction of the vector in 3-dimensional space. They are used to calculate the angle between two vectors by finding the dot product and dividing by the product of the magnitudes.

Can the angle of a vector be negative?

No, the angle of a vector is always positive. The inverse cosine function returns angles between 0 and π, so the angle of a vector will always be a positive value in radians.

How do you use the given formula to find the angle of the vector h: 3ax+5ay-8az?

To find the angle of the vector h, first calculate the dot product of the vector h with itself. This will give you the product of the magnitudes, which is equal to the squared magnitude of the vector. Then, take the square root of this value to find the magnitude of the vector. Finally, plug the values into the formula θ = cos^-1 (v1⋅v2 / ||v1|| ||v2||) to find the angle in radians.

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