adam199 said:Consider the vectors a=<2,4,-3> and b=<4,-5,6>. Determine vectors v and w such that a=v+w and v is parallel to b while w is orthogonal to b.The dot product of two orthogonal vectors is zero and the cross product of two parallel vectors is zero. A parallel vector is a multiple of the chosen vector.I tried using multiples of b for v and then seeing if random vectors orthogonal to b can be added to v to give a. I'm lost here.
Dick said:Instead of expressing parallel as a cross product, if v is parallel to b then v must be a scalar multiple of b. So v=tb for some t. That means w=a-v=a-tb. Now w.b must be 0. Try to solve for t.
adam199 said:I tried using w=a-tb. I dotted both sides by b, and got 0=(a-tb).b, where t is the only unknown, but I got stuck again. I'm not quite sure how to solve for t at that point.
Dick said:Distribute the dot product. (a-tb).b=a.b-t(b.b)=0. Now try it.
Orthogonal vectors are two vectors that are perpendicular to each other, meaning they form a 90 degree angle. Parallel vectors, on the other hand, have the same direction and magnitude, but may have different starting points.
To determine if two vectors are orthogonal, you can use the dot product. If the dot product of the two vectors is zero, then they are orthogonal. This is because the dot product is equal to the product of the magnitudes of the two vectors multiplied by the cosine of the angle between them, and if the angle is 90 degrees, the cosine will be zero.
Yes, you can have any number of orthogonal vectors in a set. As long as each vector is perpendicular to all other vectors in the set, they are considered orthogonal.
Orthogonal vectors are used in many areas of science and engineering, such as physics, computer graphics, and navigation. In physics, they are used to represent forces and motion in different directions. In computer graphics, they are used to create 3D images and animations. In navigation, they are used to determine the direction and distance of objects.
To find the components of a vector that are parallel and orthogonal to another vector, you can use the projection formula. The parallel component is equal to the dot product of the two vectors divided by the magnitude of the second vector, multiplied by the second vector. The orthogonal component is equal to the original vector minus the parallel component.