Let A be the point of intersection of the lines L1:1x−7=0y−5=−1z−3, and L2:3x−1=4y+3=5z+7. Let B and C be points on L1 and L2 respectively such that AB=AC=15. Then the square of the area of triangle ABC is:
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Step-by-Step Solution
Step 1: Find the intersection point A
First, we express the parametric equations for both lines L1 and L2. For L1, we set 1x−7=0y−5=−1z−3=t, which gives x=7+t, y=5, and z=3−t. For L2, we set 3x−1=4y+3=5z+7=s, which gives x=1+3s, y=−3+4s, and z=−7+5s. To find the intersection point A, we equate the corresponding coordinates.
Step 2: Solve for parameters and point A
Equating the y -coordinates, we get 5=−3+4s, which simplifies to 4s=8, so s=2. Substitute s=2 into the equations for x and z. From 7+t=1+3s, we get 7+t=1+3(2)=7, so t=0. From 3−t=−7+5s, we get 3−t=−7+5(2)=3, so t=0. Both equations yield t=0. Thus, the intersection point A is found by substituting t=0 into L1 's equations or s=2 into L2 's equations.
Step 3: Calculate coordinates of A
Using t=0 in the parametric equations for L1, we find the coordinates of point A: x=7+0=7, y=5, z=3−0=3. So, A=(7,5,3).
Step 4: Find direction vectors of L1 and L2
The direction vector for L1 is d1=⟨1,0,−1⟩ and for L2 is d2=⟨3,4,5⟩. These vectors represent the directions along which points B and C lie from A respectively.
Step 5: Calculate vectors AB and AC
Points B and C are on L1 and L2 respectively, such that AB=AC=15. We can find the vectors AB and AC by scaling the direction vectors d1 and d2 by the given length. The magnitude of d1 is 12+02+(−1)2=2. The magnitude of d2 is 32+42+52=9+16+25=50.
Step 6: Calculate the square of the area of triangle ABC
The square of the area of triangle ABC can be calculated using the formula Area2=41∣AB×AC∣2. First, we compute the cross product of AB and AC. We can choose the positive signs for the scaled vectors. The cross product ⟨1,0,−1⟩×⟨3,4,5⟩=⟨(0)(5)−(−1)(4),(−1)(3)−(1)(5),(1)(4)−(0)(3)⟩=⟨4,−8,4⟩. Then, we multiply by the scalar factor 25015=10015=1015=23. So, AB×AC=23⟨4,−8,4⟩=⟨6,−12,6⟩. Finally, we calculate the magnitude squared of this vector and divide by 4.