New 11.2 Vectors
The vector in component form is ⟨𝑎, 𝑏⟩ is the vector starting at (0, 0) and going through the point (a, b).
The magnitude of the vector |𝑣| is defined as √𝑎 2 + 𝑏 2 .
The vector can also be written with magnitude and direction angle like this:
(𝑟𝑐𝑜𝑠𝜃, 𝑟𝑠𝑖𝑛𝜃) where 𝑟 is the magnitude of the vector and 𝜃 is the directional angle which is the smallest angle formed by the x-axis and the vector.
If the position at any time is (𝑥(𝑡), 𝑦(𝑡)) , then the position vector is ⟨𝑥(𝑡), 𝑦(𝑡)⟩ the velocity vector is ⟨ 𝑑𝑥 𝑑𝑡
, 𝑑𝑦 𝑑𝑡
⟩ the particle’s speed is the magnitude of v denoted by |𝑣| = √( 𝑑𝑥 𝑑𝑡
)
2
+ ( 𝑑𝑦
) 𝑑𝑡
2 the particle’s acceleration vector is 𝑎(𝑡) = ⟨ 𝑑 𝑑𝑡
2 𝑥
2
, 𝑑 2 𝑑𝑡 2 𝑦
⟩
If the particle is moving along a path so that its velocity is 𝑣(𝑡) = (𝑣
1
(𝑡), 𝑣
2
(𝑡)) the displacement from 𝑡 = 𝑎 𝑡𝑜 𝑡 = 𝑏 is given by the vector ⟨∫ 𝑣
1
(𝑡)𝑑𝑡, ∫ 𝑣
2
(𝑡) ⟩ if the particle’s original position is (𝑥(𝑎), 𝑦(𝑎)) the new position would be
⟨ 𝑥(𝑎) + ∫ 𝑣
1
(𝑡)𝑑𝑡, 𝑦(𝑎) + ∫ 𝑣
2
(𝑡) ⟩ the total distance traveled is 𝑎 𝑏 𝑎 𝑏
∫ |𝑣(𝑡)|𝑑𝑡 = ∫ √(𝑣
1
(𝑡)) 2 + (𝑣
2
(𝑡)) 2 𝑑𝑡

1) A particle moves in the xy-plane so that at any time t, the position of the particle is given by 𝑥(𝑡) = 𝑡 3 + 4𝑡 2 , 𝑦(𝑡) = 𝑡 4 − 𝑡 3 a) Find the velocity vector when 𝑡 = 1.
b) Write the equation of the tangent line to the graph when 𝑡 = 1.
c) Find the acceleration vector when 𝑡 = 1.
2) A particle moves in the xy-plane so that at any time t, 𝑡 ≥ 0, the position of the particle is given by 𝑥(𝑡) = 𝑡 2 + 3𝑡, 𝑦(𝑡) = 𝑡 3 − 3𝑡 2
. Find the magnitude of the velocity vector when 𝑡 = 1.
(Find the speed of the particle.)

3) A particle moves in the xy-plane so that 𝑥 = √3 − 4𝑐𝑜𝑠𝑡 and 𝑦 = 1 − 2𝑠𝑖𝑛𝑡 , where 0 ≤ 𝑡 ≤ 2𝜋 . The path of the particle intersects the x-axis twice. Write an expression that represents the distance traveled by the particle between the two x-intercepts. Do not evaluate.
4) A particle moves in the xy-plane so that at any time t, the position of the particle is given by 𝑥(𝑡) = 2𝑡 3 − 15𝑡 2 + 36𝑡 + 5 and 𝑦(𝑡) = 𝑡 3 − 3𝑡 2 + 1 , where 𝑡 ≥ 0.
For what value(s) of t is the particle at rest?
5) No Calculator
A particle moves in the xy-plane in such a way that its velocity vector is
⟨3𝑡 2 − 4𝑡, 8𝑡 3 + 5⟩ . If the position vector at 𝑡 = 0 is ⟨7, −4⟩ , find the position of the particle at 𝑡 = 1.

6) Calculator
An object moving along a curve in the xy-plane has position (𝑥(𝑡), 𝑦(𝑡)) at time 𝑡 with 𝑑𝑥 𝑑𝑡
= sin (𝑡 3 ) and 𝑑𝑦 𝑑𝑡
= cos(𝑡 2 ) . At time 𝑡 = 2 , the object is at the position (7, 4). a) Write the equation of the tangent line to the curve at the point where 𝑡 = 4.
b) Find the speed of the vector at 𝑡 = 2.
c) For what value of t, 0 < 𝑡 < 1 , does the tangent line to the curve have a slope of 4?
Find the acceleration vector at this time. d) Find the position of the particle at time 𝑡 = 1.

1973 No calculator
1) A particle moves on the curve 𝑦 = 𝑙𝑛𝑥 so that the x-component has velocity 𝑥 ′ (𝑡) = 𝑡 + 1 for 𝑡 ≥ 0 . At time 𝑡 = 0 , the particle is at the point (1, 0) . At time 𝑡 = 1, the particle is what point?
1985
2) A particle moves in the xy-plane so that at any time t its coordinates are 𝑥 = 𝑡 2 𝑦 = 𝑡 4 − 2𝑡 3
. At 𝑡 = 1 , its acceleration vector is
− 1 and
1985
3) If the velocity of a particle moving along the x-axis is 𝑣(𝑡) = 2𝑡 − 4 and if at 𝑡 = 0 its position is 4, then at any time t its position 𝑥(𝑡) is
1988
4) For any time 𝑡 ≥ 0 , if the position of a particle in the xy-plane is given by 𝑥 = 𝑡 2 + 1 and 𝑦 = ln (2𝑡 + 1) , then the acceleration vector is

1993
5) The position of the particle moving along the x-axis is 𝑥(𝑡) = sin(2𝑡) − cos (3𝑡) for time 𝑡 ≥ 0 . When 𝑡 = 𝜋 , the acceleration of the particle is
1993
6) A particle moves along the x-axis so that at any time 𝑡 ≥ 0 , the acceleration of the particle is 𝑎(𝑡) = 𝑒 −2𝑡
. If at 𝑡 = 0 , the velocity of theparticle is
5
2
and its position is
17
, then its position at any time 𝑡 > 0 is 𝑥(𝑡) =
4
1993
7) If a particle is moves in the xy-plane so that at any time 𝑡 > 0 , its position vector is
(ln(𝑡 2 + 2𝑡) , 2𝑡 2 ) , then at time 𝑡 = 2 , its velocity vector is

1997
8) If 𝑥 = 𝑒 2𝑡
and 𝑦 = sin (2𝑡) , then 𝑑𝑦 𝑑𝑥
=
1997
9) The length of the path described by the parametric equations 𝑥 = cos 3 for 0 ≤ 𝑡 ≤ 𝜋
2
is given by what integral? 𝑡 and 𝑦 = sin 3 𝑡 ,
1997
10) For what values of 𝑡 does the curve given by parametric equations 𝑥 = 𝑡 3 𝑦 = 𝑡 4 + 2𝑡 2 − 8𝑡 have a vertical tangent?
− 𝑡 2 − 1 and
1998
11) In the xy-plane, the graph of the parametric equations 𝑥 = 5𝑡 + 2 and 𝑦 = 3𝑡 , for −3 ≤ 𝑡 ≤ 3 , is a line segment with slope

1998
12) A particle moves on a plane curve so that at any time 𝑡 > 0 its x-coordinate is 𝑡 3 its y-coordinate is (2𝑡 − 1) 3
. The acceleration vector of the particle at 𝑡 = 1 is
− 𝑡 and
1998
13) The length of the path described by the parametric equations 𝑥 =
1
3 𝑡 3
and 𝑦 = where 0 ≤ 𝑡 ≤ 1 , is given by the integral
1
2 𝑡 2
,
1998
14) If 𝑓 is the vector-valued function defined by 𝑓(𝑡) = (𝑒 −𝑡 , 𝑐𝑜𝑠𝑡) , then 𝑓 ′′ (𝑡) =

2003 No Calculator
15) For 0 ≤ 𝑡 ≤ 13 , an object travels along an elliptical path given by the parametric equations 𝑥 = 3𝑐𝑜𝑠𝑡 and 𝑦 = 4𝑠𝑖𝑛𝑡 .
At the point where 𝑡 = 13, the object leaves that path and travels along the line tangent to the path at that point. What is the slope of the line on which the object travels?
2003
16) The position of a particle moving in the xy-plane is given by the parametric equations 𝑥 = 𝑡 3 − 3𝑡 2
and 𝑦 = 2𝑡 3 − 3𝑡 2 − 12𝑡 . For what values of 𝑡 is the particle at rest?
2003
17) A curve C is defined by the parametric equations 𝑥 = 𝑡 2 − 4𝑡 + 1 and 𝑦 = 𝑡 3
. What is the equation of the line tangent to the graph of C at the point (-3, 8)?