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:
Given points are A(0, 4, 1), B(2,3,– 1), C(4, 5, 0) and D(2,6,2)
D’ratios of AB = 2,–1 –2
And d’ratios of DC = 2,–1,–2
∴ AB||DC
Similarly, d’ratios of AD = 2, 2, 1 and d’ratios of BC = 2, 2, 1
∴ AD || BC
So ABCD is a parallelogram
`vec"AB" = 2hat"i" - hat"j" - 2hat"k"`
`vec"AD" = 2hat"i" + 2hat"j" + hat"k"`
∴ Area of parallelogram ABCD = `|vec"AB" xx vec"AD"|`
= `|(hat"i", hat"j", hat"k"),(2, -1, -2),(2, 2, 1)|`
= `hat"i"(-1 + 4) - hat"j"(2 + 4) + hat"k"(4 + 2)`
= `3hat"i" - 6hat"j" + 6hat"k"`
= `sqrt((3)^2 + (-6)^2 + (6)^2)`
= `sqrt(9 + 36 + 36)`
= `sqrt(81)`
= 9 sq.units
APPEARS IN
संबंधित प्रश्न
Write the direction ratios of the following line :
`x = −3, (y−4)/3 =( 2 −z)/1`
Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane `vecr.(hati + 2hatj -5hatk) + 9 = 0`
Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
Find the angle between the lines whose direction cosines are given by the equations
(i) l + m + n = 0 and l2 + m2 − n2 = 0
Find the angle between the lines whose direction cosines are given by the equations
2l − m + 2n = 0 and mn + nl + lm = 0
Find the angle between the lines whose direction cosines are given by the equations
l + 2m + 3n = 0 and 3lm − 4ln + mn = 0
What are the direction cosines of Y-axis?
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.
Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.
Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.
Write direction cosines of a line parallel to z-axis.
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.
For every point P (x, y, z) on the xy-plane,
For every point P (x, y, z) on the x-axis (except the origin),
If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, −2) is 4, then its z-coordinate is
The direction ratios of the line which is perpendicular to the lines with direction ratios –1, 2, 2 and 0, 2, 1 are _______.
Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) .
Find the direction cosines of a vector whose direction ratios are
0, 0, 7
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 4hat"j" + 8hat"k"`
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
If a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.
If a line has the direction ratio – 18, 12, – 4, then what are its direction cosine.
The Cartesian equation of a line AB is: `(2x - 1)/2 = (y + 2)/2 = (z - 3)/3`. Find the direction cosines of a line parallel to line AB.
A line passes through the points (6, –7, –1) and (2, –3, 1). The direction cosines of the line so directed that the angle made by it with positive direction of x-axis is acute, are ______.
If two straight lines whose direction cosines are given by the relations l + m – n = 0, 3l2 + m2 + cnl = 0 are parallel, then the positive value of c is ______.
If a line makes an angle α, β and γ with positive direction of the coordinate axes, then the value of sin2α + sin2β + sin2γ will be ______.
Which identity must be satisfied by the direction cosines \[(l,m,n)\] of a line?
Any three numbers proportional to the direction cosines of a line are called its:
Which proportion correctly expresses the relationship between direction ratios \[(a,b,c)\] and direction cosines \[(l,m,n)\]?
For direction ratios \[(a,b,c)\], which set can represent the corresponding direction cosines when the direction of the line is chosen?
