Question

If A = (1, -2, -1), B = (4, 0, -3), C = (1, 2, -1) and D = (2, -4, -5), find the distance between AB and CD.

Answer 1

Answer:

Distance AB =√17 units

CD =√53 units

Step-by-step explanation:

Distance between two points can be calculated by two ways

1) vector form: by making position vector of each point,subtract both and taking magnitude

A(1,-2,-1)

Position vecotr of point A $\vec a=\hat i-2\hat j-\hat k$

B(4,0,-3)

Position vecotr of point B $\vec b=4\hat i-3\hat k$

Distance between vector a and b

$\vec a-\vec b=\hat i-2\hat j-\hat k-4\hat i+3\hat k\\\\=-3\hat i-2\hat j+2\hat k\\\\\\ magnitude\:of\: \vec a-\vec b=\sqrt{(-3)^{2}+(-2)^{2}+2^{2}   } \\\\=\sqrt{9+4+4} \\\\AB =\sqrt{17} \:\:units$

2) Cartesian form:

Distance between two points with co-ordinatesC(x1,y1,z1) D(x2,y2,z2)

$=\sqrt{(x_{2}-x_{1})^{2}+ (y_{2}-y_{1})^{2}+ (z_{2}-z_{1})^{2}}\\\\C(1,2,-1)\\\\D(2,-4,-5)\\\\=\sqrt{(2-1)^{2}+ (-4-2)^{2}+ (-5+1)^{2}}\\\\=\sqrt{1+36+16} \\\\CD=\sqrt{53}\:units$

Answer 2

the shortest distance is 4/3 units