Direction vector to a rotation (three.js)

Question

I'm attempting rotate an arrow on the surface of a planet to face the direction it is traveling. I can get the direction vector and the up vector from the surface normal. How would I go about turning this into a quaternion for my arrows rotation. I'm using three.js so any suggested methods they provided that I can use would be useful. Cheers.

The information I have so far.

var upVector = new THREE.Vector3();
upVector.subVectors( new THREE.Vector3(0,0,0), lastVec );

var dirVector = new THREE.Vector3();
upVector.subVectors( lastVec, destVec );

var sideVector = new THREE.Vector3();
sideVector.crossVectors( dirVector, upVector );

var newUp = new THREE.Vector3();
newUp.crossVectors( dirVector, sideVector );

Answer 1

Try (with myDirectionVector):

var mx = new THREE.Matrix4().lookAt(myDirectionVector,new THREE.Vector3(0,0,0),new THREE.Vector3(0,1,0));
var qt = new THREE.Quaternion().setFromRotationMatrix(mx);