Introduction to Three Dimensional Geometry
ICSE · Class 11 · Mathematics
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A point P has coordinates (2, 4, 6). Find the foot of the perpendicular from P on the xz-plane.
A point P has coordinates (-3, 5, 7). Find the reflection of P in the yz-plane and then state the new coordinates.
Two points are A(2, 3, 5) and B(8, 9, 11). Find the lengths of the edges of the rectangular parallelopiped formed by planes through A and B parallel to the coordinate planes.
Three points are A(3, 6, 9), B(6, 12, 18) and C(9, 18, 27). Use the distance formula to decide whether they are collinear.
Sample Questions
Find the point on the x-axis that is equidistant from A(1, 2, 2) and B(7, 5, 8).
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(11/2, 0, 0)
Let the point be P(x, 0, 0). Then PA = PB. So √((x-1)^2 + (0-2)^2 + (0-2)^2) = √((x-7)^2 + (0-5)^2 + (0-8)^2). Squaring gives (x-1)^2 + 8 = (x-7)^2 + 89. Simplifying: x^2 - 2x + 9 = x^2 - 14x + 138, so 12x = 129 and x = 11/2. Therefore P = (11/2, 0, 0).
A point lies in the yz-plane and is equidistant from A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1). Find the point.
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(0, 1, 3)
Let P = (0, y, z). Equating PA and PB gives 3y - z = 0. Equating PB and PC gives 3y + z = 6. Solving these two equations gives y = 1 and z = 3. So the point is (0, 1, 3).
Find the coordinates of the centroid of triangle with vertices A(1, 2, 3), B(4, 5, 6) and C(7, 8, 9).
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(4, 5, 6)
The centroid is the average of the three vertices. So G = ((1+4+7)/3, (2+5+8)/3, (3+6+9)/3) = (4, 5, 6).
Find the coordinates of the point that divides the line segment joining P(2, -3, 4) and Q(8, 0, 10) in the ratio 1:2 internally.
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(4, -2, 6)
Use the internal section formula with m:n = 1:2. Coordinates are ((1·8 + 2·2)/3, (1·0 + 2·(-3))/3, (1·10 + 2·4)/3) = (4, -2, 6).
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