English

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. - Mathematics

Advertisements
Advertisements

Question

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.

Sum
Advertisements

Solution

\[ \text { Since a unit vector makes an angle of } \frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text { with } \hat {j}  \text{ and an acute angle }  \theta \text{ with } \hat{k}  , l = \cos \frac{\pi}{3} \text { or } \frac{1}{2}, m = \cos \frac{\pi}{4}\text { or } \frac{1}{\sqrt{2}} \text { and } n = \cos \theta . \]

\[\text{ We know } \]

\[ l^2 + m^2 + n^2 = 1\]

\[ \Rightarrow \left( \frac{1}{2} \right)^2 + \left( \frac{1}{\sqrt{2}} \right)^2 + \cos^2 \theta = 1\]

\[ \Rightarrow \frac{1}{4} + \frac{1}{2} + \cos^2 \theta = 1 \]

\[ \Rightarrow \cos^2 \theta = \frac{1}{4}\]

\[ \Rightarrow \cos \theta = \frac{1}{2} \]

\[ \Rightarrow \theta = \frac{\pi}{3}\]

\[\text { Thus, the vector }  \vec{a} \text { makes an angle of } \frac{\pi}{3} \text { with }  \hat {k}  .\]

shaalaa.com
  Is there an error in this question or solution?
Chapter 27: Direction Cosines and Direction Ratios - Very Short Answers [Page 25]

APPEARS IN

RD Sharma Mathematics [English] Class 12
Chapter 27 Direction Cosines and Direction Ratios
Very Short Answers | Q 19 | Page 25

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 a line makes angles 90°, 135°, 45° with the X, Y, and Z axes respectively, then its direction cosines are _______.

(A) `0,1/sqrt2,-1/sqrt2`

(B) `0,-1/sqrt2,-1/sqrt2`

(C) `1,1/sqrt2,1/sqrt2`

(D) `0,-1/sqrt2,1/sqrt2`


Find the Direction Cosines of the Sides of the triangle Whose Vertices Are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).


If l1m1n1 and l2m2n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 − m2n1n1l2 − n2l1l1m2 ­− l2m1.


If the lines `(x-1)/(-3) = (y -2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z -6)/(-5)` are perpendicular, find the value of k.


If a line has direction ratios 2, −1, −2, determine 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 angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.


Define direction cosines of a directed line.


What are the direction cosines of Y-axis?


Write the distances of the point (7, −2, 3) from XYYZ and XZ-planes.


Write the distance of the point (3, −5, 12) from X-axis?


If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.


Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.


Find the distance of the point (2, 3, 4) from the x-axis.


For every point P (xyz) on the x-axis (except the origin),


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


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) . 


If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.


Find the direction cosines of a vector whose direction ratios are
1, 2, 3


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.


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 ______.


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")`


Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.


O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.


If the directions cosines of a line are k,k,k, then ______.


The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.


Find the direction cosine of a line which makes equal angle with coordinate axes.


What will be the value of 'P' so that the lines `(1 - x)/3 = (7y - 14)/(2P) = (z - 3)/2` and `(7 - 7x)/(3P) = (y - 5)/1 = (6 - z)/5` at right angles.


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 angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.


Find the coordinates of the image of the point (1, 6, 3) with respect to the line `vecr = (hatj + 2hatk) + λ(hati + 2hatj + 3hatk)`; where 'λ' is a scalar. Also, find the distance of the image from the y – axis.


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 ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×