MHB How to solve the following matrix rotation

[Logan Land]

Find the standard matrix for the rotation of 60◦ about the axis determined by the vector v = (3, 4, 5).

do I multiply each x,y,z by pi/3?

doesnt seem like it should be that simple

 

[Logan Land]

ok so I figured out that there is a rotational matrix in basic form that is

cos -sin
sin cos

so a rotation by 60degrees gives me a rotational matrix of
1/2 -sqrt3/2
sqrt3/2 1/2

now from here where would I proceed multiply it by v=(3,4,5)?

 

[Nono713]

The matrix you gave is for 2D vectors. Here is a hint: find a transformation that sends (3, 4, 5) to some standard axis (say, the x-axis, i.e. to a vector of the form (t, 0, 0)) then rotate the x-axis, or whichever axis you picked, 60 degrees (that can be done using the 2D matrix you gave: for standard axes, the 3D rotation matrix is similar to the 2D one, as one axis is left fixed). Then transform back. Can you come up with a basis that has (3, 4, 5) as a basis vector? What would be the change of basis matrix? Hence, what would the rotation matrix be?

 

[Logan Land]

The matrix you gave is for 2D vectors. Here is a hint: find a transformation that sends (3, 4, 5) to some standard axis (say, the x-axis, i.e. to a vector of the form (t, 0, 0)) then rotate the x-axis, or whichever axis you picked, 60 degrees (that can be done using the 2D matrix you gave: for standard axes, the 3D rotation matrix is similar to the 2D one, as one axis is left fixed). Then transform back. Can you come up with a basis that has (3, 4, 5) as a basis vector? What would be the change of basis matrix? Hence, what would the rotation matrix be?

im confused
i don't no how to get 3,4,5 to t,0,0 or 0,t,0 or 0,0,t
could i get an example then i can try and figure out my problem

can i use a rotational matrix that's 3d? and not have to perform a transformation?

 

[Nono713]

im confused
i don't no how to get 3,4,5 to t,0,0 or 0,t,0 or 0,0,t
could i get an example then i can try and figure out my problem

can i use a rotational matrix that's 3d? and not have to perform a transformation?

Have you learned change of basis matrices? What are you expected to use to solve this problem?

 

[Logan Land]

Have you learned change of basis matrices? What are you expected to use to solve this problem?

we only have 1 vector set though... (3,4,5)
what would I use to obtain another basis?
We've always been given multiple vectors