Advertisements
Advertisements
Question
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
Advertisements
Solution
\[\text{ Let } \vec{a} \text{ be a vector with direction ratios } 2, 3, - 6 . \]
\[ \Rightarrow \vec{a} = 2 \hat{i} + 3 \hat{j} - 6 \hat{k} .\]
\[\ \text { Let } \vec{b} \text { be a vector with direction ratios } 3, - 4, 5 . \]
\[ \Rightarrow \vec{b} = 3 \hat{i} - 4 \hat{j} + 5 \hat{k} \]
\[\text{ Let } \theta \text{ be the angle between the given vectors } . \]
\[\text{ Now, }\]
\[\cos \theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} \]
\[ = \frac{\left( 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) . \left( 3 \hat{i} - 4 \hat{j} + 5 \hat{k} \right)}{\left| 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right|\left| 3 \hat{i} - 4 \hat{j} + 5 \hat{k} \right|}\]
\[ = \frac{6 - 12 - 30}{\sqrt{4 + 9 + 36} \sqrt{9 + 16 + 25}} \]
\[ = \frac{- 36}{\sqrt{49} \sqrt{50}} \]
\[ = \frac{- 36}{35\sqrt{2}}\]
\[\text{ Rationalising the result, we get }\]
\[\cos \theta = - \frac{18\sqrt{2}}{35} \]
\[ \therefore \theta = \cos^{- 1} \left( - \frac{18\sqrt{2}}{35} \right)\]
\[\ \text { Thus, the angle between the given vectors measures }\cos^{- 1} \left( - \frac{18\sqrt{2}}{35} \right) . \]
APPEARS IN
RELATED QUESTIONS
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.
Find the Direction Cosines of the Sides of the triangle Whose Vertices Are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).
If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 − m2n1, n1l2 − n2l1, l1m2 − l2m1.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.
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?
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
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
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 xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
The angle between the two diagonals of a cube 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 direction cosines of a vector whose direction ratios are
`1/sqrt(2), 1/2, 1/2`
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
`3hat"i" - 3hat"k" + 4hat"j"`
Find the direction cosines and direction ratios for the following vector
`hat"i" - hat"k"`
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 `vec"a", vec"b", vec"c"`
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"`
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 `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines 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 ______.
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")`
If a line has the direction ratio – 18, 12, – 4, then what are its direction cosine.
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 ______.
Equation of line passing through origin and making 30°, 60° and 90° with x, y, z axes respectively, is ______.
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.
A directed line makes angles \[\alpha\], \[\beta\], and \[\gamma\] with the positive x-, y-, and z-axes respectively. What are these angles called?
If \[\alpha\], \[\beta\], and \[\gamma\] are the direction angles of a line, which ordered triple gives its direction cosines?
Which proportion correctly expresses the relationship between direction ratios \[(a,b,c)\] and direction cosines \[(l,m,n)\]?
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 \[PQ\]?
For the line directed from \[P(x_1,y_1,z_1)\] to \[Q(x_2,y_2,z_2)\], which expression is the direction cosine \[m\]?
