Advertisements
Advertisements
प्रश्न
A vector `vec"r"` has magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and components of `vec"r"`, given that `vec"r"` makes an acute angle with x-axis.
Advertisements
उत्तर
Let `vec"a", vec"b"` and `vec"c"` three vectors such that `vec"a" = 2"k", vec"b"` = 3k and `vec"c"` = – 6k
If l, m and n are the direction cosines of vector `vec'r"`, then
l = `vec"a"/|vec"r"| = (2"k")/14 = "k"/7`
m = `vec"b"/|vec"r"| = (3"k")/14` and n = `vec"c"/|vec"r"| = (-6"k")/14 = (-3"k")/7`
We know that l2 + m2 + n2 = 1
∴ `"k"^2/49 + (9"k"^2)/196 + (9"k"^2)/49` = 1
⇒ `(4"k"^2 + 9"k"^2 + 36"k"^2)/196` = 1
⇒ 49k2 = 196
⇒ k2 = 4
∴ k = ± 2 and l = `"k"/7 = 2/7`
m = `(3"k")/14 = (3 xx 2)/14 = 3/7`
And n = `(-3"k")/7 (-3 xx 2)/7 = (-6)/7`
∴ `hat"r" = +- (2/7hat"i" + 3/7hat"j" - 6/7hat"k")`
`hat"r" = hat"r"|vec"r"|`
⇒ `vec"r" = +-(2/7hat"i" + 3/7hat"j" - 6/7hat"k")*14`
= `+- (4hat"i" + 6hat"j" - 12hat"k")`
Hence, the required direction cosines are `2/7, 3/7, (-6)/7` and the components of `vec"r"` are `4hat"i", 6hat"j"` and `-12hat"k"`.
APPEARS IN
संबंधित प्रश्न
Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector
`2hati+3hatj+4hatk` to the plane `vecr` . `(2hati+hatj+3hatk)−26=0` . Also find image of P in the plane.
Find the position vector of a point which divides the join of points with position vectors `veca-2vecb" and "2veca+vecb`externally in the ratio 2 : 1
Find the value of 'p' for which the vectors `3hati+2hatj+9hatk and hati-2phatj+3hatk` are parallel
If `bara, barb, barc` are position vectors of the points A, B, C respectively such that `3bara+ 5barb-8barc = 0`, find the ratio in which A divides BC.
Classify the following as scalar and vector quantity.
Time period
In Figure, identify the following vector.
Coinitial
Two vectors having the same magnitude are collinear.
Find the direction cosines of the vector joining the points A (1, 2, -3) and B (-1, -2, 1) directed from A to B.
Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).
If θ is the angle between two vectors `veca` and `vecb`, then `veca . vecb >= 0` only when ______.
Express \[\vec{AB}\] in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (4, −1), B (1, 3)
Find \[\left| \vec{A} B \right|\] in each case.
Express \[\vec{AB}\] in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (−6, 3), B (−2, −5)
Find \[\left| \vec{A} B \right|\] in each case.
Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\] \[\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} \text{ and } \vec{b} = 4\hat{i} + 4 \hat{j} - 2\hat{k}\]
Find the angle between the vectors \[\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} + \hat{j} - 2 \hat{k}\]
Find the angle between the vectors \[\vec{a} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{b} = \hat{i} - \hat{j} + \hat{k}\]
Find the angles which the vector \[\vec{a} = \hat{i} -\hat {j} + \sqrt{2} \hat{k}\] makes with the coordinate axes.
The adjacent sides of a parallelogram are represented by the vectors \[\vec{a} = \hat{i} + \hat{j} - \hat{k}\text{ and }\vec{b} = - 2 \hat{i} + \hat{j} + 2 \hat{k} .\]
Find unit vectors parallel to the diagonals of the parallelogram.
If \[\hat{ a } \text{ and } \hat{b }\] are unit vectors inclined at an angle θ, prove that
\[\tan\frac{\theta}{2} = \frac{\left| \hat{a} -\hat{b} \right|}{\left| \hat{a} + \hat{b} \right|}\]
If \[\left| \vec{a} + \vec{b} \right| = 60, \left| \vec{a} - \vec{b} \right| = 40 \text{ and } \left| \vec{b} \right| = 46, \text{ find } \left| \vec{a} \right|\]
If \[\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}\] \[\vec{b} = \hat{i} + \hat{j} - 2 \hat{k}\] \[\vec{c} = \hat{i} + 3 \hat{j} - \hat{k}\] find λ such that \[\vec{a}\] is perpendicular to \[\lambda \vec{b} + \vec{c}\]
If \[\vec{p} = 5 \hat{i} + \lambda \hat{j} - 3 \hat{k} \text{ and } \vec{q} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,\] then find the value of λ, so that \[\vec{p} + \vec{q}\] and \[\vec{p} - \vec{q}\] are perpendicular vectors.
If \[\vec{\alpha} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \text{ and } \vec{\beta} = 2 \hat{i} + \hat{j} - 4 \hat{k} ,\] then express \[\vec{\beta}\] in the form of \[\vec{\beta} = \vec{\beta_1} + \vec{\beta_2} ,\] where \[\vec{\beta_1}\] is parallel to \[\vec{\alpha} \text{ and } \vec{\beta_2}\] is perpendicular to \[\vec{\alpha}\]
If either \[\vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0}\] then \[\vec{a} \cdot \vec{b} = 0 .\] But the converse need not be true. Justify your answer with an example.
Show that the vectors \[\vec{a} = 3 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} - 3 \hat{j} + 5 \hat{k} , \vec{c} = 2 \hat{i} + \hat{j} - 4 \hat{k}\] form a right-angled triangle.
Find the angles of a triangle whose vertices are A (0, −1, −2), B (3, 1, 4) and C (5, 7, 1).
Find the magnitude of two vectors \[\vec{a} \text{ and } \vec{b}\] that are of the same magnitude, are inclined at 60° and whose scalar product is 1/2.
If A, B and C have position vectors (0, 1, 1), (3, 1, 5) and (0, 3, 3) respectively, show that ∆ ABC is right-angled at C.
If \[\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} = 2 \hat{i} - \hat{j} + 3 \hat{k} \text{ and }\vec{c} = \hat{i} - 2 \hat{j} + \hat{k} ,\] find a unit vector parallel to \[2 \vec{a} - \vec{b} + 3 \vec{c .}\]
If \[\vec{a} \times \vec{b} = \vec{c} \times \vec{d} \text { and } \vec{a} \times \vec{c} = \vec{b} \times \vec{d}\] , show that \[\vec{a} - \vec{d}\] is parallel to \[\vec{b} - \vec{c}\] where \[\vec{a} \neq \vec{d} \text { and } \vec{b} \neq \vec{c}\] .
Find the sine of the angle between the vectors `vec"a" = 3hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" - 2hat"j" + 4hat"k"`.
If A, B, C, D are the points with position vectors `hat"i" + hat"j" - hat"k", 2hat"i" - hat"j" + 3hat"k", 2hat"i" - 3hat"k", 3hat"i" - 2hat"j" + hat"k"`, respectively, find the projection of `vec"AB"` along `vec"CD"`.
The altitude through vertex C of a triangle ABC, with position vectors of vertices `veca, vecb, vecc` respectively is:
If points A, B and C have position vectors `2hati, hatj` and `2hatk` respectively, then show that ΔABC is an isosceles triangle.
