Dot and Cross Products of Vectors in Space

Dot Product to Find Orthogonal Vectors

100

Find the dot product of u and v. Then, determine if u and v are orthogonal. 

u = (3,-9,6) v = (-8,2,7)

u multiplied by v does not equal 0.

Not orthogonal.

200

u = (5,0,-4) v = (6,-1,4)

u and v do not equal 0

Not orthogonal 

300

u = (2,-8,-7) v = (5,9,-7)

Not orthoganal 

400

u = (-7,-3,1) v = (-4,5,-13)

Not orthoganal

500

u = (11,4,-2) v = (-1,3,8)

Not orthognanal 

Angle Between Two Vectors in Space

100

Find the angle θ between vectors u and v to the nearest tenth of a degree. 

u = (3,-2,2) v = (-8,-9,5)

cos^-1(-19/sqrt17688) or 98.2 degrees

200

u = (6,-5,1) v = (-8,-9,5)

acos(2/sqrt10540) or 88.8 degrees

300

u = (-8,1,12) v = (-6,4,2)

acos(76/sqrt11704) or 45.3 degrees

400

u = (10,0,-8) v = (3,-1,-12)

acos(126/sqrt252526) or 37.5 degrees

500

u = (11,4,-2) v = (-1,3,8) 

acos(-15/sqrt10434) or 98.4 degrees