Vectors AM = (x - 1, y - 1) and U = (2, -5) are parallel if and only ifĮxpand and simplify to obtain the equation of the lineī) A point M(x, y) is on the line through point B(-2, -3) and perpendicular to vector U = (2, -5) if and only if the vectors BM and U are perpendicular. Let us first find the components of vectors AM. MB = 0 is written using the components of the two vectorsĮxpand and simplify to obtain the equation of the circleĪ) the equation of the line through point A(1, 1) and parallel to vector U.ī) the equation of the line through point B(-2, -3) and perpendicular to vector U.Ī) A point M(x, y) is on the line through point A(1, 1) and parallel to vector U = (2, -5) if and only if the vectors AM and U are parallel.Let us first find the components of vectors MA and MB given the coordinates of the three points. Triangle AMC is right at point M if and only if the scalar product MA BC = 0 is written using the components of the two vectorsįind the equation of the circle with diameter the points A(2, -2) and B(4, -3).įor a point M(x, y) to be on the circle defined by its diameter, triangle AMC must be a right triangle with the right angle at M.Let us first find the components of vectors BA and BC given the coordinates of the three points.īC = (2 k - 2, -4 - 3 ) = (2 k - 2, -7) And two vectors are perpendicular if and only if their scalar product is equal to zero. Rewrite the above condition using the components of vectors, we obtain the equationĮxpand and rearrange to obtain the quadratic equationįind the real number k so that the points A(-2, k), B(2, 3) and C(2k, -4) are the vertices of a right triangle with right angle at B.ĪBC is a right triangle at B if and only if vectors BA and BC are perpendicular. The condition for two vectors A = (Ax, Ay) and B = ( Bx, By) to be perpendicular is: Ax Bx + Ay By=0 Vectors B and C are not parallel (there no need to test since A and B are parallel)įind the real number a so that the vectors A = (2a, 16) and B = (3a+2, -2) are perpendicular The condition for two vectors A = (Ax, Ay) and B = ( Bx, By) to be parallel is: Ax By = Bx Ay.
Hence vectors A and B are perpendicular if and only if
Vectors A and B are perpendicular if and only if A Two vectors A and B are perpendicular if and only if their scalar product is equal to zero. ( Ax, Ay) = k ( Bx, By) = ( k Ax, k By)Īx = k Bx and Ay = k By or Ax / Bx = k and Ay / By = kĬondition under which vectors A = ( Ax, Ay) and B = ( Bx, By) are parallel is given by Two vectors A and B are parallel if and only if they are scalar multiples of one another.Ī = k B, k is a constant not equal to zero.Ī and B are parallel if and only if A = k B