Question

You only receive 3 attempts at this problem.

let be the subspace of all vectors [ x ]
[ y ]
[ z ]
with x and y real. for each vector below, determine whether is orthogonal to all vectors in W.
[-8]
V = [-9]
[-2]

Answer 1

For vectors A and B to be orthogonal in the xy-plane and have the same magnitude, the components of B must ensure that their dot product with A equals zero. Two solutions for B would be -4.0Î + 3.0ĸ or 4.0Î - 3.0ĸ. In three dimensions, a third unit vector Êk is included for the z-axis in a right-handed Cartesian coordinate system.

Step-by-step explanation:

To solve the provided vector problem, where vectors A and B are orthogonal vectors in the xy-plane and have identical magnitudes, we must first understand that if one vector is represented as à = 3.0Π+ 4.0ĸ, the orthogonal vector B must have components that when dotted with A, result in zero (since orthogonal vectors have a dot product of zero). Since A has components 3 and 4, B could have components -4 and 3 or 4 and -3 to ensure that their dot product is zero. Therefore, B could be either -4.0Π+ 3.0ĸ or 4.0Π- 3.0ĸ. This is because the dot product (3)(-4)+(4)(3) = 0 or (3)(4)+(4)(-3) = 0, respectively, showing that both pairs of vectors are orthogonal.

In a Cartesian coordinate system, it's important to use orthogonal unit vectors like Î, ĸ, and Êk for the x, y, and z axes, respectively. This applies not only to two-dimensions but extends to three dimensions where the z-axis comes into play, using a right-handed coordinate system for defining orientation.

In the context of the original problem, the student likely refers to a subspace W which wasn't fully defined, but the mentioned vector V = [-8, -9, -2] would be orthogonal to vectors in W if the dot product with any vector in W results in zero.