A point is on the x-axis. What are its y-coordinate and z-coordinates?
y and z - coordinates are zero
A point is in the XZ-plane. What can you say about its y-coordinate?
y - coordinate is zero
Name the octants in which the following points lie:
(1, 2, 3), (4, −2, 3), (4, −2, −5), (4, 2, −5), (−4, 2, −5), (−4, 2, 5), (−3, −1, 6), (−2, −4, −7)
I, IV, VIII, V, VI, II, III, VII
Fill in the blanks:
(i) The x-axis and y-axis taken together determine a plane known as ______.
(ii) The coordinates of points in the XY-plane are of the form ______.
(iii) Coordinate planes divide the space into _____ octants.
(i) XY - plane
(ii) (x, y, 0)
(iii) Eight
Find the distance between the following pairs of points:
(i) \((2, 3, 5)\) and \((4, 3, 1)\)
(ii) \((-3, 7, 2)\) and \((2, 4, -1)\)
(iii) \((-1, 3, -4)\) and \((1, -3, 4)\)
(iv) \((2, -1, 3)\) and \((-2, 1, 3)\)
(i) \(2\sqrt{5}\)
(ii) \(\sqrt{43}\)
(iii) \(2\sqrt{26}\)
(iv) \(2\sqrt{5}\)
Show that the points \((-2, 3, 5)\), \((1, 2, 3)\) and \((7, 0, -1)\) are collinear.
Verify the following:
(i) \((0, 7, -10), (1, 6, -6)\) and \((4, 9, -6)\) are the vertices of an isosceles triangle.
(ii) \((0, 7, 10), (-1, 6, 6)\) and \((-4, 9, 6)\) are the vertices of a right angled triangle.
(iii) \((-1, 2, 1), (1, -2, 5), (4, -7, 8)\) and \((2, -3, 4)\) are the vertices of a parallelogram.
Find the equation of the set of points which are equidistant from the points \((1, 2, 3)\) and \((3, 2, -1)\).
\(x - 2z = 0\)
Find the equation of the set of points \(P\), the sum of whose distances from \(A(4, 0, 0)\) and \(B(-4, 0, 0)\) is equal to 10.
\(9x^2 + 25y^2 + 25z^2 - 225 = 0\)
Three vertices of a parallelogram ABCD are A(3, −1, 2), B(1, 2, −4) and C(−1, 1, 2). Find the coordinates of the fourth vertex.
(1, −2, 8)
Find the lengths of the medians of the triangle with vertices A(0, 0, 6), B(0, 4, 0) and C(6, 0, 0).
7, \(\sqrt{34}\), 7
If the origin is the centroid of the triangle PQR with vertices P(2a, 2, 6), Q(−4, 3b, −10) and R(8, 14, 2c), find the values of \(a\), \(b\) and \(c\).
\(a = -2\), \(b = -\dfrac{16}{3}\), \(c = 2\)
If A and B be the points (3, 4, 5) and (−1, 3, −7), respectively, find the equation of the set of points P such that \(PA^2 + PB^2 = k^2\), where \(k\) is a constant.
\(x^2 + y^2 + z^2 - 2x - 7y + 2z = \dfrac{k^2 - 109}{2}\)