5. Let V = R3 and let (, ·) be the standard dot prod- uct. (a) Calculate (3e, + 2e, – e3, 2e, + 4e3). (b) Let W Span(e, + e, + e3) and find W+. (c) Using the Gram Schmidt process, find an or- thogonal basis for V containing the vector ej – Eg + 2e3. |

Explain the Combination of basis vectors.

Please don't provide handwritten solution ......

Make the vectors u1=(2,0,-1,1), u2=(1,1,-1,3), u3= (-1,1,1,-1) orthonormal.

Two soccer players are practicing for an upcoming game. One of them runs 10m from point A to point B. She then turns left at 90 degrees and runs 10m until she reaches point C. Then she kicks the ball with a speed of 10 m/s at an upward angle of 30 degrees to her teammate, who is located at point A. Write the initial velocity vector of the ball in component form, explicitly describing the basis vectors you are using.

Decompose the vector ¯b = ⟨1, 0, 4⟩ into its projection proja¯(¯b) and its orthogonal component orthoa¯(¯b), where ¯a = ⟨1, 1, 1⟩.

Find a vector perpendicular to (2, −1, 10) and (1, 2, −1)