can someone explain to me how i can find the following:
Hyperplane H in R6 is perpendicular to the vector (1, -1, 2, 1, -1, 0) and contains both the point P(a, a, a, a, a, a) an point Q(a, -a, -a, a+2, 1, 3) for some specific value a
What is the value of a?
thank so much
Hyperplane H in R6 is perpendicular to the vector (1, -1, 2, 1, -1, 0) and contains both the point P(a, a, a, a, a, a) an point Q(a, -a, -a, a+2, 1, 3) for some specific value a
What is the value of a?
thank so much
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If P and Q are on the hyperplane, then the vector QP is parallel to the plane.
QP = OP - OQ = (a, -a, -a, a+2, 1, 3) - (a, a, a, a, a, a) = (0, -2a, -2a, 2, 1-a, 3-a)
Since the vector (1, -1, 2, 1, -1, 0) is perpendicular to the plane, then
(1, -1, 2, 1, -1, 0) • (0, -2a, -2a, 2, 1-a, 3-a) = 0
2a - 4a + 2 -1 + a = 0
1 - a = 0
a = 1
QP = OP - OQ = (a, -a, -a, a+2, 1, 3) - (a, a, a, a, a, a) = (0, -2a, -2a, 2, 1-a, 3-a)
Since the vector (1, -1, 2, 1, -1, 0) is perpendicular to the plane, then
(1, -1, 2, 1, -1, 0) • (0, -2a, -2a, 2, 1-a, 3-a) = 0
2a - 4a + 2 -1 + a = 0
1 - a = 0
a = 1