BT ⊥ MV
The direction vector BT is ...
... T - B = (5, 12) - (1, -4) = (4, 16)
The direction vector MV is ...
... V - M = (-4, 2) - (-8, 3) = (4, -1)
One is not a multiple of the other, so the lines are not parallel.
The dot-product of these direction vectors is
... (4, 16)•(4, -1) = 4·4 + 16·(-1) = 0
When the dot-product is zero, the vectors are perpendicular.
the lines are perpendicular
to determine which case is true we require the slope m of the lines
Parallel lines have equal slopes
Perpendicular slopes are the negative inverse of each other
to calculate m use the gradient formula
m = ( y₂ - y₁ ) / ( x₂ - x₁ )
with (x₁, y₁ ) = B(1, - 4 ) and (x₂, y₂ ) = T(5, 12 )
[tex]m_{BT}[/tex] = [tex]\frac{12+4}{5-1}[/tex] = [tex]\frac{16}{4}[/tex] = 4
repeat with
(x₁, y₁ ) = M(-8, 3 ) and (x₂, y₂ ) = V(-4, 2 )
[tex]m_{MV}[/tex] = [tex]\frac{2-3}{-4+8}[/tex] = - [tex]\frac{1}{4}[/tex]
4 and - [tex]\frac{1}{4}[/tex] are negative inverses, hence
BT and MV are perpendicular to each other
