Advertisements
Advertisements
Question
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
Advertisements
Solution
\[ \text{ Let the YZ - plane divide the line segment joining points } P\left( - 2, 5, 9 \right) \text { and Q } \left( 3, - 2, 4 \right) \text{ in the ratio k: 1 } . \]
\[ \text{ Using the section formula, the coordinates of the point of intersection are given by }\]
\[\left( \frac{k\left( 3 \right) - 2}{k + 1}, \frac{k\left( - 2 \right) + 5}{k + 1}, \frac{k\left( 4 \right) + 9}{k + 1} \right)\]
\[ \text{ On the YZ - plane, the X - coordinate of any point is zero } . \]
\[ \therefore \frac{k\left( 3 \right) - 2}{k + 1} = 0\]
\[ \Rightarrow 3k - 2 = 0\]
\[ \Rightarrow k = \frac{2}{3}\]
\[ \text{ Thus, the YZ - plane divides the line segment formed by joining the given points in the ratio 2: 3 internally } . \]
APPEARS IN
RELATED QUESTIONS
Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1
If l, m, n are the direction cosines of a line, then prove that l2 + m2 + n2 = 1. Hence find the
direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.
If a line has the direction ratios −18, 12, −4, then what are its direction cosines?
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
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 lines whose direction cosines are given by the equations
l + 2m + 3n = 0 and 3lm − 4ln + mn = 0
What are the direction cosines of X-axis?
What are the direction cosines of Y-axis?
What are the direction cosines of Z-axis?
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the distance of the point (3, −5, 12) from X-axis?
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 inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
If a unit vector `vec a` makes an angle \[\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}\] and an acute angle θ with \[\hat{ k} \] ,then find the value of θ.
A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is
If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
If O is the origin, OP = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of P are
The angle between the two diagonals of a cube is
Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) .
Verify whether the following ratios are direction cosines of some vector or not
`4/3, 0, 3/4`
Find the direction cosines and direction ratios for the following vector
`3hat"i" + hat"j" + hat"k"`
Find the direction cosines and direction ratios for the following vector
`hat"j"`
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 3hat"k" + 4hat"j"`
A triangle is formed by joining the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). Find the direction cosines of the medians
If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`, find the magnitude and direction cosines of `3vec"a"- 2vec"b"+ 5vec"c"`
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.
If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.
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 ______.
If the directions cosines of a line are k,k,k, then ______.
If a line has the direction ratio – 18, 12, – 4, then what are its direction cosine.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.
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:
A line passes through \[P(x_1,y_1,z_1)\] and \[Q(x_2,y_2,z_2)\]. Which is one set of direction ratios of the line?
For points \[P(x_1,y_1,z_1)\] and \[Q(x_2,y_2,z_2)\], which expression gives \[PQ\]?
