For the points
A(1,−9,1),B(−8,−7,−7) and C(−8,3,−7)
find the position vector t of the point T on AB such that
vector AT=−5 x vector TB.
t = (3,0,16) but I'm not sure how to approach this problem. Any help is very much appreciated
To find the position vector of point T on AB such that vector AT is equal to -5 times vector TB, you can use the concept of vector addition and scalar multiplication.
First, let's find vector AB (the displacement vector from A to B):
AB = B - A = (-8, -7, -7) - (1, -9, 1) = (-9, 2, -8)
Now, let's express vector AT in terms of vector TB:
AT = -5 * TB
Since vector AT is in the same direction as vector AB, we can write:
AT = k * AB (where k is a scalar)
Substituting the values, we have:
-5 * TB = k * (-9, 2, -8)
Comparing the corresponding components, we get the following equations:
-5 * TBx = -9k
-5 * TBy = 2k
-5 * TBz = -8k
Now, we have a system of equations to solve for the scalar k.
From the second equation, we can find k as:
k = -5 * TBy / 2
Similarly, from the third equation:
k = -5 * TBz / -8
Setting these two expressions for k equal to each other, we have:
-5 * TBy / 2 = -5 * TBz / -8
Simplifying, we get:
TBy = 4 * TBz
Substituting this back into the second equation:
-5 * (4 * TBz) = 2k
Simplifying further:
-20 * TBz = 2k
Now, we can express TBz in terms of a parameter t:
TBz = t
Substituting this back into the equation, we get:
-20t = 2k
Simplifying again:
k = -10t
Now, we have expressions for both TBy and k in terms of t:
TBy = 4 * t
k = -10t
Finally, we can express the position vector t of point T on AB:
T = A + t * AB
Substituting the values:
T = (1, -9, 1) + t * (-9, 2, -8)
T = (1 - 9t, -9 + 2t, 1 - 8t)
Therefore, the position vector of point T on AB such that vector AT is -5 times vector TB is given by:
t = (1 - 9t, -9 + 2t, 1 - 8t)
In this specific case, when substituting t = 3 into the equation, the position vector t of point T is:
t = (1 - 9(3), -9 + 2(3), 1 - 8(3))
t = (1 - 27, -9 + 6, 1 - 24)
t = (-26, -3, -23)
So, the position vector t for point T is (or, in more precise terms, the coordinates of point T are) (3, 0, 16).