Advertisements
Advertisements
प्रश्न
Prove that the point A(1, 3, 0), B(–5, 5, 2), C(–9, –1, 2) and D(–3, –3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.
उत्तर
Let A(1,3,0),B( \[-\]5,5,2), C(\[-\]9,\[-\]1,2) and D(\[-\]3,\[-\]3,0) be the coordinates of quadrilateral \[\square ABCD\]
\[AB = \sqrt{\left( - 5 - 1 \right)^2 + \left( 5 - 3 \right)^2 + \left( 2 - 0 \right)^2}\]
\[ = \sqrt{36 + 4 + 4}\]
\[ = \sqrt{44}\]
\[BC = \sqrt{\left( - 9 + 5 \right)^2 + \left( - 1 - 5 \right)^2 + \left( 2 - 2 \right)^2}\]
\[ = \sqrt{16 + 36 + 0}\]
\[ = \sqrt{52} \]
\[CD = \sqrt{\left( - 3 + 9 \right)^2 + \left( - 3 + 1 \right)^2 + \left( 0 - 2 \right)^2}\]
\[ = \sqrt{36 + 4 + 4}\]
\[ = \sqrt{44}\]
\[DA = \sqrt{\left( 1 + 3 \right)^2 + \left( 3 + 3 \right)^2 + \left( 0 - 0 \right)^2}\]
\[ = \sqrt{16 + 36 + 0}\]
\[ = \sqrt{52}\]
Here, we see that AB = CD& BC = DA
Since, opposite pair of sides are equal .
Therefore,
\[\square ABCD\] is a parallelogram .
\[AC = \sqrt{\left( - 9 - 1 \right)^2 + \left( - 1 - 3 \right)^2 + \left( 2 - 0 \right)^2}\]
\[ = \sqrt{\left( - 10 \right)^2 + \left( - 4 \right)^2 + \left( 2 \right)^2}\]
\[ = \sqrt{100 + 16 + 4}\]
\[ = \sqrt{120}\],m
\[BD = \sqrt{\left( - 3 + 5 \right)^2 + \left( - 3 - 5 \right)^2 + \left( 0 - 2 \right)^2}\]
\[ = \sqrt{\left( 2 \right)^2 + \left( - 8 \right)^2 + \left( - 2 \right)^2}\]
\[ = \sqrt{4 + 64 + 4}\]
\[ = \sqrt{72}\]
\[\therefore AC \neq BD\]
∴ ABCD is not a rectangle.
APPEARS IN
संबंधित प्रश्न
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.
Name the octants in which the following points lie: (5, 2, 3)
Name the octants in which the following points lie:
(–5, 4, 3)
Name the octants in which the following points lie:
(7, 4, –3)
Name the octants in which the following points lie:
(–5, –3, –2)
Name the octants in which the following points lie:
(–7, 2 – 5)
Find the image of:
(5, 2, –7) in the xy-plane.
Find the image of:
(–5, 0, 3) in the xz-plane.
A cube of side 5 has one vertex at the point (1, 0, –1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the coordinates of the other vertices of the cube.
Find the distances of the point P(–4, 3, 5) from the coordinate axes.
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Show that the points A(1, 2, 3), B(–1, –2, –1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram ABCD, but not a rectangle.
Write the distance of the point P (2, 3,5) from the xy-plane.
XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
The coordinates of the foot of the perpendicular drawn from the point P(3, 4, 5) on the yz- plane are
The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are
The perpendicular distance of the point P (6, 7, 8) from xy - plane is
The length of the perpendicular drawn from the point P (3, 4, 5) on y-axis is
If a line makes an angle of 30°, 60°, 90° with the positive direction of x, y, z-axes, respectively, then find its direction cosines.
The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.
Find the co-ordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, –1, 3) and C(2, –3, –1).
Find the image of the point having position vector `hati + 3hatj + 4hatk` in the plane `hatr * (2hati - hatj + hatk)` + 3 = 0.
If the line drawn from the point (–2, – 1, – 3) meets a plane at right angle at the point (1, – 3, 3), find the equation of the plane
Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0
If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ2 = δl2 + δm2 + δn2
Two systems of rectangular axis have the same origin. If a plane cuts them at distances a, b, c and a′, b′, c′, respectively, from the origin, prove that
`1/a^2 + 1/b^2 + 1/c^2 = 1/(a"'"^2) + 1/(b"'"^2) + 1/(c"'"^2)`
Find the foot of perpendicular from the point (2,3,–8) to the line `(4 - x)/2 = y/6 = (1 - z)/3`. Also, find the perpendicular distance from the given point to the line.
Find the length and the foot of perpendicular from the point `(1, 3/2, 2)` to the plane 2x – 2y + 4z + 5 = 0.
The direction cosines of the vector `(2hati + 2hatj - hatk)` are ______.
The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
The intercepts made by the plane 2x – 3y + 5z +4 = 0 on the co-ordinate axis are `-2, 4/3, - 4/5`.
If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2), then the equation of plane is `vecr.(5hati - 3hatj - 2hatk)` = 38.
