Advertisements
Advertisements
प्रश्न
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
उत्तर
\[\text{Let}\ \vec{a}\ \text{be a vector with direction ratios}\ 1, - 2, 1 . \]
\[ \Rightarrow \vec{a} =\hat{ i} - 2 \hat{j} + \hat {k} . \]
\[\text{Let} \ \vec{b}\ \text{be a vector with direction ratios} \ 4, 3, 2 . \]
\[ \Rightarrow \vec{b} = 4\hat{ i} + 3 \hat{j} + 2 \hat{k} . \]
\[\text{ Let }\ \theta \text{ be the angle between the given vectors } . \]
\[\text{ Now, } \]
\[\text{ cos }\theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} \]
\[ = \frac{\left( \hat{ i } - 2\hat { j } + \ \hat{k} \right) . \left( 4\hat { i } + 3 \ \hat{j}+ 2 \ \hat {k} \right)}{\left| \hat { i } - 2 \ \hat { j } +\ \hat {k} \right|\left| 4 \ \hat { i }+ 3\ \hat { j } + 2 \ \hat { k } \right|}\]
\[ = \frac{4 - 6 + 2}{\sqrt{1 + 4 + 1} \sqrt{16 + 9 + 4}} \]
\[ = \frac{0}{\sqrt{6} \sqrt{29}} \]
\[ = 0 \]
\[ \therefore \theta = \frac{\pi}{2}\]
\[\text{Thus, the angle between the given vectors measures }\frac{\pi}{2} .\]
APPEARS IN
संबंधित प्रश्न
Write the direction ratios of the following line :
`x = −3, (y−4)/3 =( 2 −z)/1`
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
If a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find 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 acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.
Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.
If the coordinates of the points A, B, C, D are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between AB and CD.
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
Define direction cosines of a directed line.
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.
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.
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.
For every point P (x, y, z) on the x-axis (except the origin),
A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is
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 distance of the point P (a, b, c) from the x-axis is
Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
Verify whether the following ratios are direction cosines of some vector or not
`1/sqrt(2), 1/2, 1/2`
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
`5hat"i" - 3hat"j" - 48hat"k"`
If (a, a + b, a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, 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.
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 ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
The vector equation of the line passing through the points (3, 5, 4) and (5, 8, 11) is `vec"r" = 3hat"i" + 5hat"j" + 4hat"k" + lambda(2hat"i" + 3hat"j" + 7hat"k")`
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 ______.
The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
If a line makes angles 90°, 135°, 45° with x, y and z-axis respectively then which of the following will be its direction cosine.
If a line has the direction ratio – 18, 12, – 4, then what are its direction cosine.
The d.c's of a line whose direction ratios are 2, 3, –6, are ______.
If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.
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 ______.
