Advertisements
Advertisements
प्रश्न
Using distance formula prove that the following points are collinear:
A(3, –5, 1), B(–1, 0, 8) and C(7, –10, –6)
Advertisements
उत्तर
AB =\[\sqrt{\left( - 1 - 3 \right)^2 + \left( 0 + 5 \right)^2 + \left( 8 - 1 \right)^2}\]
\[= \sqrt{\left( - 4 \right)^2 + \left( 5 \right)^2 + \left( 7 \right)^2}\]
\[ = \sqrt{16 + 25 + 49}\]
\[ = \sqrt{90}\]
\[ = 3\sqrt{10}\]
BC =\[\sqrt{\left( 7 + 1 \right)^2 + \left( - 10 - 0 \right)^2 + \left( - 6 - 8 \right)^2}\]
\[= \sqrt{\left( 8 \right)^2 + \left( - 10 \right)^2 + \left( - 14 \right)^2}\]
\[ = \sqrt{64 + 100 + 196}\]
\[ = \sqrt{360}\]
\[ = 6\sqrt{10}\]
AC =\[\sqrt{\left( 7 - 3 \right)^2 + \left( - 10 + 5 \right)^2 + \left( - 6 - 1 \right)^2}\]
\[= \sqrt{\left( 4 \right)^2 + \left( - 5 \right)^2 + \left( - 7 \right)^2}\]
\[ = \sqrt{16 + 25 + 49}\]
\[ = \sqrt{90}\]
\[ = 3\sqrt{10}\]
\[Here, AB + AC = 3\sqrt{10} + 3\sqrt{10}\]
\[ = 6\sqrt{10}\]
\[ = BC\]
Hence, the points are collinear.
APPEARS IN
संबंधित प्रश्न
Find the distance between the following pairs of points:
(–1, 3, –4) and (1, –3, 4)
Find the distance between the following pairs of points:
(2, –1, 3) and (–2, 1, 3)
Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.
Verify the following:
(0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of a right angled triangle.
Verify the following:
(–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 P, the sum of whose distances from A (4, 0, 0) and B (–4, 0, 0) is equal to 10.
Find the distance between the following pairs of point:
A(3, 2, –1) and B(–1, –1, –1).
Find the distance between the points P and Q having coordinates (–2, 3, 1) and (2, 1, 2).
Using distance formula prove that the following points are collinear:
A(4, –3, –1), B(5, –7, 6) and C(3, 1, –8)
Using distance formula prove that the following points are collinear:
P(0, 7, –7), Q(1, 4, –5) and R(–1, 10, –9)
Show that the points (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of an isosceles right-angled triangle.
Prove that the tetrahedron with vertices at the points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.
Find the centroid of a triangle, mid-points of whose sides are (1, 2, –3), (3, 0, 1) and (–1, 1, –4).
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, –6) respectively, find the coordinates of the point C.
If the distance between the points P(a, 2, 1) and Q (1, −1, 1) is 5 units, find the value of a.
Find the distance of the point whose position vector is `(2hati + hatj - hatk)` from the plane `vecr * (hati - 2hatj + 4hatk)` = 9
Find the distance of the point (– 2, 4, – 5) from the line `(x + 3)/3 = (y - 4)/5 = (z + 8)/6`
The distance of a point P(a, b, c) from x-axis is ______.
Prove that the line through A(0, –1, –1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(– 4, 4, 4).
Find the equation of a plane which is at a distance `3sqrt(3)` units from origin and the normal to which is equally inclined to coordinate axis
Find the distance of a point (2, 4, –1) from the line `(x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)`
Find the shortest distance between the lines given by `vecr = (8 + 3lambdahati - (9 + 16lambda)hatj + (10 + 7lambda)hatk` and `vecr = 15hati + 29hatj + 5hatk + mu(3hati + 8hatj - 5hatk)`
Distance of the point (α, β, γ) from y-axis is ______.
If one of the diameters of the circle x2 + y2 – 2x – 6y + 6 = 0 is a chord of another circle 'C' whose center is at (2, 1), then its radius is ______.
