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Chapter 11. Vector Functions
ยง11.1. Vector Functions and Space Curves
1. Find the limit lim
๎ธ โ โ
๎ ๎ธ ๎ ๎ญ
๎ต๎ธ
๎ถ ๎ ๎ด
๎ถ ๎ ๎ธ
๎ ๎ธ sin ๎ญ ๎ธ
- Draw the projections of the curve ๎ถ ๎๎ธ๎
๎^ โฉ๎ธ๎ sin ๎ธ๎ ๎ตcos ๎ธโช on the three coordinate planes. Use
these projections to help sketch the curve.
3. Find a vector equation and parametric equations for the line segment that joins ๎ ๎๎ฝ๎ ๎ ๎ด๎ ๎ด ๎
^ and
- Find a vector function the represents the curve of intersection of the cylinder ๎ผ
๎ต
๎ ๎ฝ
๎ต
๎ ๎ท and the
surface ๎พ ๎ ๎ผ๎ฝ.
- Two particles travel along the space curves ๎ถ ๎ด
๎ต ๎ ๎ธ
๎ถ
โช and ๎ถ
๎ต
๎๎ธ๎
๎ (^) โฉ๎ด ๎ ๎ต๎ธ๎ ๎ด ๎ ๎น๎ธ๎ ๎ด ๎ ๎ด๎ท๎ธโช Do
the particles collide? Do their paths intersect?
ยง11.2. Derivatives and Integrals of Vector Functions
1. Find the unit tangent vector ๎ ๎๎ธ๎
^ of ๎ถ ๎๎ธ๎
๎^ โฉcos ๎ธ๎ ๎ถ๎ธ๎ ๎ต sin ๎ต ๎ธโช at ๎ธ ๎ ๎ฝ.
- Find parametric equations for the tangent line to the curve with parametric equations ๎ผ ๎ ๎ฉ
๎ ๎ธ
cos ๎ธ,
๎ ๎ธ
sin ๎ธ, ๎พ ๎ ๎ฉ
๎ ๎ธ
at the point ๎๎ด๎ ๎ฝ๎ ๎ด๎
.
3. Find the point on the curve ๎ถ ๎๎ธ๎
๎^ โฉ๎ต cos ๎ธ๎ ๎ต sin ๎ธ๎ ๎ฉ
๎ธ
โช, ๎ฝ โค ๎ธ โค ๎ฌ, where the tangent line is parallel
to the plane ๎
- (a) The curves ๎ถ ๎ด
๎ต
๎ ๎ธ
๎ถ
โช and ๎ถ
๎ต
๎๎ธ๎
๎ โฉsin ๎ธ๎ sin ๎ต๎ธ๎ ๎ธโช intersect at the origin. Find their angle
of intersection.
(b) At what point do the curves ๎ถ ๎ด
๎ต
โช and ๎ถ
๎ต
๎ต
โช intersect? Find
their angle of intersection.
- Evaluate the integral ๎
๎ด
๎ต
๎ต
๎ญ ๎ ๎ธ๎^
๎ธ ๎ ๎ด ๎ฎ ๎ ๎ธ sin ๎ฌ๎ธ๎ฏ๎
๎จ๎ธ.
6. Find ๎ชโฒ๎๎ต๎
, where ๎ช ๎๎ธ๎
๎ ๎น๎๎ธ๎
โ ๎บ๎๎ธ๎
, ๎น๎๎ต๎
๎ โฉ๎ด๎ ๎ต๎ ๎ ๎ดโช, ๎นโฒ ๎๎ต๎
๎ โฉ๎ถ๎ ๎ฝ๎ ๎ทโช, and ๎บ๎๎ธ๎
๎ โฉ๎ธ๎ ๎ธ
๎ต ๎ ๎ธ
๎ถ
- Show that if ๎ถ is a vector function such that ๎ถโณ exists, then ๎ญ ๎จ๎ธ
๎ ๎ถ ๎๎ธ๎
ร ๎ถโฒ๎๎ธ๎
๎ ๎ ๎ถ ๎๎ธ๎
ร ๎ถโณ๎๎ธ๎
.
- If ๎ถ ๎๎ธ๎
โ ๎ฝ, show that ๎ญ ๎จ๎ธ
- If a curve has the property that the position vector ๎ถ ๎๎ธ๎
is always perpendicular to the tangent
vector ๎ถโฒ๎๎ธ๎
, show that the curve lies on a sphere with center the origin๎ where ๎ถ ๎๎ธ๎
โ ๎ฝ ๎
- Determine whether the following statements are true or false. If it is true, explain why.
Otherwise, explain why or give an example that disproves the statement.
(a) If ๎ถ๎๎ธ๎
โ ๎ฝ, then ๎ญ ๎จ๎ธ
๎ต ๎ ๎ต๎๎ถ๎๎ธ๎
๎ ๎๎ถโฒ๎๎ธ๎
๎.
(b) If ๎๎ถ๎๎ธ๎
๎๎ ๎ด for all ๎ธ, then ๎๎ถโฒ๎๎ธ๎
๎^ is constant.
