MTH 234 calculus 3 Exam 1 (Michigan State University) spring semester Latest Version 2026/2027|Questions and Correct Answers (Verified, Exam of mathematics

MTH 234 CALCULUS 3 EXAM 1 (MICHIGAN STATE
UNIVERSITY) SPRING SEMESTER LATEST VERSION 03/06/2026
2026/2027




MTH 234 calculus 3 Exam 1 (Michigan State University) spring
semester Latest Version 2026/2027|Questions and Correct
Answers (Verified, Exam of mathematics


distance between three points

√(x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²




-Sphere with radius r

-Center point of a sphere

-(x-x₀)²+(y-y₀)²+(z-z₀)²=r²

-(x₀,y₀,z₀)




complete the square help

(x-a)²=x²-2ax+a²




Vector AB→

B-A




dot product formula using cos of u and v (u.v)
3/6/2026




|u|²*|v|²*cos(α)


1

, MTH 234 CALCULUS 3 EXAM 1 (MICHIGAN STATE
UNIVERSITY) SPRING SEMESTER LATEST VERSION 03/06/2026
2026/2027




(u.v)

u₁*v₁+u₂*v₂+u₃*v₃




α=

cos⁻¹(u*v/(|u|*|v|))




projuV=

(u*v)/|u|²*u




compuV=

(u*v)/|u|




orthuV=

V-projuV




UxV

(u₂v₃-u₃v₂)i-(u₁v₃-u₃v₁)j+(u₁v₂-u₂v₁)k
3/6/2026




axa=


2