How do you prove two straight lines intersect?

There are two ways to prove they intercept (you can choose whichever one you prefer). Say we have two straight lines, for example: r₁ = 3i + 4j - 5k + t (i - 2j + 2k)      =>      r₁ = (3 + t)i + (4 – 2t)j + (2t – 5)k r₂ = 9i + j - 2k + s (4i + j - k)     =>    r₂ = (9 + 4s)i + (1 + s)j + (-2 -s)k The first method makes use of the fact that r gives you a position vector along the straight line. We want to find the point where the two straight lines, r₁ and r₂, give the same position vector as this is the point they intercept. To do this we need to find a value for s and t which makes r₁ = r₂. This is done by equating the i components of r₁ and r₂ and also the j components. This creates a simultaneous equation which we can solve for t and s. These values are checked by equating the k components and substituting either s or t into the equation. If it gives the same value as before then the lines intercept, if it doesn’t then they do not. Substituting s or t back into r₁ or r₂, gives the position vector at which they intercept. The other method is using the dot product: a∙b = |a||b|cos(θ). If the two straight lines are parallel then they will never intercept. Therefore, to prove they intercept, you simply have to prove they are not parallel. Parallel lines have an angle between of 0 degrees (θ = 0). So, using that cos(0) = 1 we can show that two parallel lines follow the rule a∙b = |a||b|. If we calculate r₁∙r₂, | r₁ | and | r₂| we can use this rule to prove they are not parallel and therefore intercept.