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 ______.
पर्याय
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
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:
(2, –5, –7)
Find the image of:
(5, 2, –7) in the xy-plane.
Find the image of:
(–5, 0, 3) in the xz-plane.
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 points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
Show that the points A(3, 3, 3), B(0, 6, 3), C(1, 7, 7) and D(4, 4, 7) are the vertices of a square.
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Are the points A(3, 6, 9), B(10, 20, 30) and C(25, –41, 5), the vertices of a right-angled triangle?
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.
Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).
Show that the plane ax + by + cz + d = 0 divides the line joining the points (x1, y1, z1) and (x2, y2, z2) in the ratio \[- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}\]
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.
Write the coordinates of the foot of the perpendicular from the point (1, 2, 3) on y-axis.
Find the point on x-axis which is equidistant from the points A (3, 2, 2) and B (5, 5, 4).
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 length of the perpendicular drawn from the point P(a, b, c) from z-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.
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.
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
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).
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
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 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 equations of the line passing through the point (3,0,1) and parallel to the planes x + 2y = 0 and 3y – z = 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.
Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.
Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.
If the directions cosines of a line are k, k, k, then ______.
The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
The angle between the planes `vecr.(2hati - 3hatj + hatk)` = 1 and `vecr.(hati - hatj)` = 4 is `cos^-1((-5)/sqrt(58))`.
