Advertisements
Advertisements
प्रश्न
The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.
विकल्प
9 sq.units
18 sq.units
27 sq.units
81 sq.units
Advertisements
उत्तर
The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to 9 sq.units.
Explanation:
We have, `vec(AB) = (2 - 0)hati + (3 - 4)hatj + (-1 -1)hatk = 2hati - hatj - 2hatk`
`vec(BC) = (4 - 2)hati + (5 - 3)hatj + (0 + 1)hatk = 2hati + 2hatj + hatk`
`vec(CD) = (2 - 4)hati + (6 - 5)hatj + (2 - 0)hatk = -2hati + hatj + 2hatk`
`vec(DA) = (0 - 2)hati + (4 - 6)hatj + (1 - 2)hatk = -2hati - 2hatj - hatk`
∴ Area of quadrilateral ABCD = `|vec(AB) xx vec(BC)|`
= `|(hati, hatj, hatk),(2, -1, -2),(2, 2, 1)|`
= `|hati(-1 + 4) - hatj(2 - 4) + hatk(4 + 2)|`
= `|3hati - 6hatj + 6hatk|`
= `sqrt(9 + 36 + 36)`
= 9 sq.units
APPEARS IN
संबंधित प्रश्न
The x-axis and y-axis taken together determine a plane known as_______.
Name the octants in which the following points lie: (5, 2, 3)
Name the octants in which the following points lie:
(7, 4, –3)
Name the octants in which the following points lie:
(2, –5, –7)
Find the image of:
(–2, 3, 4) in the yz-plane.
Find the image of:
(5, 2, –7) in the xy-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.
The coordinates of a point are (3, –2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
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.
If A(–2, 2, 3) and B(13, –3, 13) are two points.
Find the locus of a point P which moves in such a way the 3PA = 2PB.
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Write the distance of the point P (2, 3,5) from the xy-plane.
The coordinates of the mid-points of sides AB, BC and CA of △ABC are D(1, 2, −3), E(3, 0,1) and F(−1, 1, −4) respectively. Write the coordinates of its centroid.
Find the ratio in which the line segment joining the points (2, 4,5) and (3, −5, 4) is divided by the yz-plane.
The ratio in which the line joining (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is
What is the locus of a point for which y = 0, z = 0?
The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are
The length of the perpendicular drawn from the point P (3, 4, 5) on y-axis is
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
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.
A plane meets the co-ordinates axis in A, B, C such that the centroid of the ∆ABC is the point (α, β, γ). Show that the equation of the plane is `x/alpha + y/beta + z/γ` = 3
If a line makes an angle of `pi/4` with each of y and z axis, then the angle which it makes with x-axis is ______.
Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 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
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
Find the length and the foot of perpendicular from the point `(1, 3/2, 2)` to the plane 2x – 2y + 4z + 5 = 0.
Find the equation of the plane through the points (2, 1, –1) and (–1, 3, 4), and perpendicular to the plane x – 2y + 4z = 10.
If the directions cosines of a line are k, k, k, then ______.
The sine of the angle between the straight line `(x - 2)/3 = (y - 3)/4 = (z - 4)/5` and the plane 2x – 2y + z = 5 is ______.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.
The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is `vecr = (5hati - 4hatj + 6hatk) + lambda(3hati + 7hatj - 2hatk)`.
