Math, asked by gopika3497, 1 year ago

25. Find the point on the line joining two points (-3,5,-6) and (-5,1,-8) where the line crosses XY - plane.​

Answers

Answered by suskumari135
1

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)

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