In the case of vectors, how do I find the shortest distance between a point and a line?

The key word in the question is 'shortest'. The shortest distance from a point to a line is always the perpendicular distance. If we calculate the appropriate angles and lengths/distances, we can use trigonometry to find the shortest distance.Example question: The line l1 is (1,-1,-7) + s(2,1,5). The line l2 is (17,4,-6) + t(4,1,-3). The position vector B is (9,2,0). B lies on the line l2. The position vector A is (5, 1, 3). A is at the point where the lines l1 and l2 intersect. Find the shortest distance from B to line l1.Step 1: Draw a diagram (use whiteboard to draw one)Step 2: Find the acute angle between l1 and l2. Use the direction vectors of the lines in the equation cos(angle) = (u).(v)/|u||v| so, cos(angle) = (2,1,5).(4,1,-3)/(2²+1²+5²)^1/2x(4²+1²+(-3)²)^1/2 hence, angle = cos^-1(-6/2(195)^1/2) = 102.4.. the acute angle = 180 - 102.4.. = 77.6 (to 1dp)Step 3: Find the direction vector between A and B, AB. AB = B - A = (9,2,0) - (5,1,3) = (4,1,-3)Step 4: Find the magnitude of AB. |AB| = (4² +1²+(-3)²)^1/2 = (26)^1/2Step 5: Find the shortest distance, D, using trigonometry. sin(angle) = D/|AB| Rearrange this equation to give, D = (26)^1/2x sin77.6 = 4.98(3sf)