23 December 2021 Intersection point of two vectors

23 December 2021 Intersection point of two vectors LO: Find the coordinates of the point of intersection of two vectors. www. mathssupport. org

Intersecting lines In a two dimensions plane, two lines will either Intersect: If the value of the parameters is consistent in the two equations Be parallel: If they have direction vectors that are scalar multiples of each other. In a three dimensions plane, two lines will either Intersect: If the value of the parameters is consistent in all three equations Be parallel: If they have direction vectors that are scalar multiples of each other. If the lines are not parallel and the values Be skew: are not consistent so the lines do not intersect. www. mathssupport. org

Intersecting point of two vectors 1 3 Two lines have equations: r 1 = 0 + s 1 -1 1 0 6 and r 2 = 2 + t 4 0 8 Show that the lines intersect and find the coordinates of the point of intersection. r 1 and r 2 intersect if there is a value of t and a value of s such that r 1 = r 2 Equating the components of each vector Solving for s and t in and Substituting s and t in 3+s=6 s = 2 + 4 t -1 + s = 8 t s=3 3 = 2 + 4 t 2=2 Since the value of s and t are consistent for all three equations the two lines must intersect. www. mathssupport. org

Intersecting point of two vectors 1 3 Two lines have equations: r 1 = 0 + s 1 -1 1 0 6 and r 2 = 2 + t 4 0 8 To find the point of intersection: Substitute the value of s into r 1 Equating the components of each vector r 1 = 1 3 x y = 0 +3 1 -1 1 z x=3+3 y=0+3 z = -1 + 3 x=6 y=3 z=2 Therefore the coordinates of the point of intersection are: (6, 3, 2) www. mathssupport. org

Intersecting point of two vectors 1 3 Two lines have equations: r 1 = 0 + s 1 -1 1 0 6 and r 2 = 2 + t 4 0 8 To find the point of intersection: We can substitute the value of t into r 2 Equating the components of each vector r 2 = 6 x y = 2 0 z x=6+0 0 4 8 x=6 y=3 z=2 This gives the same coordinates and it’s a useful way of checking the answers (6, 3, 2) www. mathssupport. org

Intersecting point of two vectors 3 Two lines have equations: r 1 = 0 5 1 1 -1 + s 1 and r 2 = 4 + t 1 0 1 2 Show that the lines are skew. r 1 and r 2 intersect if there is a value of t and a value of s such that r 1 = r 2 Equating the components of each vector Solving for s and t s+t=2 s-t=4 s=3 3 -s=1+t s=4+t 5 + 2 s = t 3 -t=4 t = -1 5 + 2(3) = -1 11 ≠ -1 Since the value of s and t are not consistent for all three equations the two lines do not intersect, they are skew. www. mathssupport. org

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