25. Find the point on the line joining two points (-3,5,-6) and (-5,1,-8) where the line crosses XY - plane.
Answers
The point on the line joining is (-1,17,0)
Step-by-step explanation:
The equation of line passing through two points with position vector a and b
vector r = a + λ(b-a)
Given the line passes through the points
A( -3, 5, -6) and B(-5, 1, -8)
a = -4iΛ +5jΛ -6 kΛ
b = -5iΛ + jΛ -8kΛ
(b-a) = (-5iΛ + jΛ -8kΛ) - (-4iΛ +5jΛ -6 kΛ)
= -5iΛ + jΛ - 8kΛ +4iΛ -5jΛ +6kΛ
= -i ∧ -4j∧ -2k∧
r = -4iΛ +5jΛ -6 kΛ + λ (-i ∧ -4j∧ -2k∧)........(i)
Let the coordinates of the point where the line crosses the XY plane be( x, y, 0)
So, r = xiΛ +yjΛ + 0kΛ......(ii)
Since the point crosses the plane , it will satisfy the equation
Put ii in i
xiΛ +yjΛ + 0kΛ = -4iΛ +5jΛ -6 kΛ + λ (-i ∧ -4j∧ -2k∧)
= (-4-1λ) i∧ + (5-4λ)j∧ + (-6-2λ)k∧
Two vectors are equal if their corresponding components are equal
So, x = -4 - λ
y = 5 -4λ
0 = -6 -2λ
-2λ = 6
λ = -3
Put λ value x and y
x = -4 -(-3) = -4 +3 = -1
y = 5 - 4(-3) = 5+12 = 17
Therefore, the required points are (-1 ,17 , 0)