If line joining points A and B having position vectors 6a 4b + 4c and 4c respectively, - Maths - Vector Algebra

Question

If line joining points A and B having position vectors 6a −
4b + 4c and −4c respectively, and the line joining the points
C and D having position vectors −a −2b −3c and a
+ 2b −5c intersect, then their point of intersection is

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

Let O be originOA→=6a→-4b→+4c→ 6,-4,4OB→=-4c→AB→=OB→-OA→=-4c→-6a→+4b→-4c→AB→=-6a→+4b→-8c→Hence direction ratios of AB are -6,4,-8 i
e
 3,-2,4Hence equation of AB isx-63=y--4-2=z-44=k⇒x-63=y+4-2=z-44=kHence any general point on AB will be x,y,z=3k+6,-2k-4,4k+4Similarly equation of CD will bex+11=y+22=z+3-1=λHence any general point on CD will be x,y,z=λ-1,2λ-2,-λ-3At point of intersection, general point of AB and CD will be same⇒x=x and y=y⇒3k+6=λ-1  and -2k-4=2λ-2⇒3k-λ=-7
i and 2λ+2k=-2 i e  λ+k=-1
iiAdding equations we get3k-λ+λ+k=-7-1⇒4k=-8⇒k=-2x,y,z=3k+6,-2k-4,4k+4=-6+6,4-4,-8+4=0,0,-4Hence they intersect at -4c→

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If line joining points A and B having position vectors 6a − 4b + 4c and −4c respectively, and the line joining the points C and D having position vectors −a −2b −3c and a + 2b −5c intersect, then their point of intersection is