Advertisements
Advertisements
प्रश्न
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
Advertisements
उत्तर
\[\text{ Let } \vec{a} \text{ be a vector with direction ratios } 2, 3, - 6 . \]
\[ \Rightarrow \vec{a} = 2 \hat{i} + 3 \hat{j} - 6 \hat{k} .\]
\[\ \text { Let } \vec{b} \text { be a vector with direction ratios } 3, - 4, 5 . \]
\[ \Rightarrow \vec{b} = 3 \hat{i} - 4 \hat{j} + 5 \hat{k} \]
\[\text{ Let } \theta \text{ be the angle between the given vectors } . \]
\[\text{ Now, }\]
\[\cos \theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} \]
\[ = \frac{\left( 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) . \left( 3 \hat{i} - 4 \hat{j} + 5 \hat{k} \right)}{\left| 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right|\left| 3 \hat{i} - 4 \hat{j} + 5 \hat{k} \right|}\]
\[ = \frac{6 - 12 - 30}{\sqrt{4 + 9 + 36} \sqrt{9 + 16 + 25}} \]
\[ = \frac{- 36}{\sqrt{49} \sqrt{50}} \]
\[ = \frac{- 36}{35\sqrt{2}}\]
\[\text{ Rationalising the result, we get }\]
\[\cos \theta = - \frac{18\sqrt{2}}{35} \]
\[ \therefore \theta = \cos^{- 1} \left( - \frac{18\sqrt{2}}{35} \right)\]
\[\ \text { Thus, the angle between the given vectors measures }\cos^{- 1} \left( - \frac{18\sqrt{2}}{35} \right) . \]
APPEARS IN
संबंधित प्रश्न
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.
Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane `vecr.(hati + 2hatj -5hatk) + 9 = 0`
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.
What are the direction cosines of Y-axis?
What are the direction cosines of Z-axis?
Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).
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.
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
Write direction cosines of a line parallel to z-axis.
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.
For every point P (x, y, z) on the xy-plane,
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
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
If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos2 α + cos2 β + cos2γ + cos2 δ is equal to
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 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
`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
`3hat"i" - 3hat"k" + 4hat"j"`
Find the direction cosines and direction ratios for the following vector
`hat"i" - hat"k"`
If `1/2, 1/sqrt(2), "a"` are the direction cosines of some vector, then find a
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"`
Choose the correct alternative:
The unit vector parallel to the resultant of the vectors `hat"i" + hat"j" - hat"k"` and `hat"i" - 2hat"j" + hat"k"` 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 ______.
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.
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 ______.
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 a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, 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.
