# Maths Vector Helpppp

The points A and B have position vectors, relative to the origin O, given by
−−→OA = i+2j+3k and −−→OB = 2i+j+3k.
The line l has vector equation
r = (1−2t)i+ (5+t)j+ (2−t)k.
(i) Show that l does not intersect the line passing through A and B. 
(ii) The point P lies on l and is such that angle PAB is equal to 60◦
. Given that the position vector
of P is (1 − 2t)i + (5 + t)j + (2 − t)k, show that 3t
2
+ 7t + 2 = 0. Hence ﬁnd the only possible
position vector of P.

I got everything else except that Idk how to find the possible vector of P.
t=-2 and -3

so how do you know which to use?

Ans 5i + 3j +4k, means they used t=-2

Thank you!

1. 👍 0
2. 👎 0
3. 👁 3,173
1. For easier typing I will use <...> to represent a vector and (.. ..)to represent a point.

I can write the line l in parametric form
x = 1-2t
y = 5+t
z = 2-t

let P(x,y,z) be the point we are looking for on line l
or P is (1-2t, 5+t, 2-t)
vector AP = <1-2t-1, 5+t-2, 2-t-3> or
vector AP = <-2t , 3+t , -1-t >
similarly
vector AB = <1,-1,0>
AB dot AP= |AB||AP|cos60°
-2t -3-t + 0 = √2 √(4t^2 + (3+t)^2 + (-1-t)^2) (1/2)
-3-3t = √2 √(4t^2 + 9+6t+t^2 + 1+2t+t^2) (1/2)
-6-6t = √2 √(6t^2 + 8t + 10) ******
square both sides
36 + 72t + 36t^2 = 12t^2 + 16t + 20
24t^2 +56t + 16 = 0
divide by 8
3t^2 + 7t + 2 = 0 -----> as requested

(t+2)(3t + 1) = 0
t = -2 or t = -1/3

check: in ******
if t = -2
LS = -6+12 = 6
RS = √2 √(24 -16+10)
= √2√18
= √36 = 6 = LS

If t + -1/3
LS = -6 + 2 = -4
But by definition of √, √(of anything) is the Positive result,
so LS ≠ RS

only t = -1 works, which gives us vector OP = <5,3,4>
and P is the point (5,3,4)

1. 👍 6
2. 👎 0

1. 👍 1
2. 👎 0

## Similar Questions

1. ### PHYSICS

Vector A⃗ points in the negative x direction and has a magnitude of 22 units. The vector B⃗ points in the positive y direction. partA:Find the magnitude of B⃗ if A⃗ +B⃗ has a magnitude of 37 units. partB: Sketch A⃗ and

2. ### Physics

A ball is lodged in a hole in the floor near the outside edge of a merry-go-round that is turning at constant speed. Which kinematic variable or variables change with time, assuming that the position is measured from an origin at

3. ### physics

the position vectors of the head and tail of radius vectors are 2i+j+k and 2i-3j+k. the linear momomentum is 2i+3j+k. the angular momentum is

4. ### Calculus

Sketch the vector field F⃗ (r⃗ )=2r⃗ in the plane, where r⃗ =⟨x,y⟩. Select all that apply. A.All the vectors point away from the origin. B. The vectors increase in length as you move away from the origin. C. All the

1. ### Physics

1) The vector sum of three vectors gives a resultant equal to zero. what can you say about the vectors? 2) Vector is 3.00 units in length and points along the positive x-axis. Vector is 4.00 units in length and points along the

2. ### Calculus

Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. a(t) = 2 i + 6t j + 12t2 k, v(0) = i, r(0) = 2 j − 7 k

3. ### math

I am looking for help to this question On the \$xy\$-plane, the origin is labeled with an \$M\$. The points \$(1,0)\$, \$(-1,0)\$, \$(0,1)\$, and \$(0,-1)\$ are labeled with \$A\$'s. The points \$(2,0)\$, \$(1,1)\$, \$(0,2)\$, \$(-1, 1)\$, \$(-2, 0)\$,

4. ### phy

A boat can travel 2.60 m/s in still water.If the boat points its prow directly across a stream whose current is 1.10 m/s, what is the velocity (magnitude and direction) of the boat relative to the shore? What will be the position

1. ### Math

The position vectors, r km, and velocity vectors, vkm/h, of two boats at a certain time are given by Boat A: 10 am, r(a) = 6i+10j, v(a) = 8i-2j Boat B: 10:30 am, r(b) = 8i+5j, v(b) = 4i+j If the boats collide, find the time when

2. ### Physics

From a velocity graph, can you tell where the starting position is? Also, can you tell where the ending position is relative to the starting position? Explain.

3. ### physics

In order to change the angular momentum of an object, it is necessary to apply a. a net force to the object. b. a net torque to the object. c. a centripetal acceleration to the object. d. both a net force and a net torque to the

4. ### vectors( need answer asap)

The centroid of the triangle OAB is denoted by G. If O is the origin and OA=4i + 3j ,OB=6i-j. Find OG in terms of the unit vectors, I and j.