Advertisements
Advertisements
प्रश्न
If l1, m1, n1 ; l2, m2, n2 ; l3, m3, n3 are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3 makes equal angles with them.
Advertisements
उत्तर
Let direction vector of three mutually perpendicular lines be
`veca = l_1hati + m_1hatj + n_1hatk`
`vecb = l_2hati + m_2hatj + n_2hatk`
`vecc = l_3hati + m_3hatj + n_3hatk`
Let the direction vector associated with directions cosines l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3 be
`vecp = (1_1 + l_2 + l_3)hati + (m_1 + m_2 + m_3)hati + (n_1 + n_2 + n_3)hatk`
As, lines associated with direction vectors a, b and c are mutually perpendicular
We have `veca . vecb` = 0 ......[dot product of two perpendicular vector is 0]
⇒ l1l2 + m1m2 + n1n2 = 0 .....[1]
Similarly, `veca . vecc` = 0
⇒ l1l3 + m1m3 + n1n3 = 0 ......[2]
And `vecb . vecc` = 0
⇒ l2l3 + m2m3 + n2n3 = 0 ......[3]
Now, let x, y, z be the angles made by direction vectors a, b and c respectively with p
Therefore, `cosx = veca. vecp`
⇒ cos x = l1(l1 + l2 + l3) + m1(m1 + m2 + m3) + n1(n1+ n2 + n3)
⇒ cos x = l12 + l1l2 + l1l3 + m12 + m1m2 + m1m3 + n12 + n1n2 + n1n3
⇒ cos x = l12 + m12 + n12 + (l1l2 + m1m2 + n1n2) + (l1l3 + m1m3 + n1n3)
As we know l12 + m12 + n12 = 1 because sum of squares of direction cosines of a line is equal to 1
⇒ cos x = 1 + 0 = 1 ......[From, 1 and 2]
Similarly, cos y = 1 and cos z = 1
⇒ x = y = z = 0
Hence, angle made by vector p, with vectors a, b and c are equal!
APPEARS IN
संबंधित प्रश्न
Name the octants in which the following points lie:
(1, 2, 3), (4, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5),
(–3, –1, 6), (2, –4, –7).
Coordinate planes divide the space into ______ octants.
Name the octants in which the following points lie:
(–5, –4, 7)
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, 4, –3) in the xz-plane.
Find the image of:
(–5, 0, 3) in the xz-plane.
Planes are drawn through the points (5, 0, 2) and (3, –2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed.
Find the distances of the point P(–4, 3, 5) from the coordinate axes.
Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.
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.
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.
Find the locus of P if PA2 + PB2 = 2k2, where A and B are the points (3, 4, 5) and (–1, 3, –7).
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 point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.
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 \[- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}\]
Write the distance of the point P(3, 4, 5) from z-axis.
Write the coordinates of the foot of the perpendicular from the point (1, 2, 3) on y-axis.
Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.
Let (3, 4, –1) and (–1, 2, 3) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to
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 perpendicular distance of the point P(3, 3,4) from the x-axis is
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
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).
The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.
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
The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle α. Prove that the equation of the plane in its new position is ax + by `+- (sqrt(a^2 + b^2) tan alpha)z ` = 0
The line `vecr = 2hati - 3hatj - hatk + lambda(hati - hatj + 2hatk)` lies in the plane `vecr.(3hati + hatj - hatk) + 2` = 0.
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.
