Advertisements
Advertisements
प्रश्न
Show that the vector `hati + hatj + hatk` is equally inclined to the axes OX, OY, and OZ.
Advertisements
उत्तर
Let `veca = hati + hatj + hatk`
Then,
`|veca| =sqrt(1^2 + 1^2 + 1^2) = sqrt3`
Therefore, the direction cosines of `veca` are `(1/sqrt3, 1/sqrt3, 1/sqrt3)`.
Now, let α, β, and λ be the angles formed by `veca` with the positive directions of the x, y, and z axes.
Then, we have `cosalpha = 1/sqrt3, cosbeta = 1/sqrt3, coslambda = 1/sqrt3`.
Hence, the given vector is equally inclined to axes OX, OY, and OZ.
APPEARS IN
संबंधित प्रश्न
Write the position vector of the point which divides the join of points with position vectors `3veca-2vecb and 2veca+3vecb` in the ratio 2 : 1.
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
Represent graphically a displacement of 40 km, 30° east of north.
`veca and -veca` are collinear.
Find the direction cosines of the vector `hati + 2hatj + 3hatk`.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `hati + 2hatj - hatk` and `-hati + hatj + hatk` respectively, externally in the ratio 2:1.
Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).
Show that the points A, B and C with position vectors `veca = 3hati - 4hatj - 4hatk`, `vecb = 2hati - hatj + hatk` and `vecc = hati - 3hatj - 5hatk`, respectively form the vertices of a right angled triangle.
Find the value of x for which `x(hati + hatj + hatk)` is a unit vector.
If θ is the angle between two vectors `veca` and `vecb`, then `veca . vecb >= 0` only when ______.
Find a vector of magnitude 4 units which is parallel to the vector \[\sqrt{3} \hat{i} + \hat{j}\]
ABCD is a parallelogram. If the coordinates of A, B, C are (−2, −1), (3, 0) and (1, −2) respectively, find the coordinates of D.
Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\] \[\vec{a} = 3\hat{i} - 2\hat{j} - 6\hat{k} \text{ and } \vec{b} = 4 \hat{i} - \hat{j} + 8 \hat{k}\]
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 \[\vec{a,} \vec{b,} \vec{c}\] are three mutually perpendicular unit vectors, then prove that \[\left| \vec{a} + \vec{b} + \vec{c} \right| = \sqrt{3}\]
Show that the vector \[\hat{i} + \hat{j} + \hat{k}\] is equally inclined to the coordinate axes.
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.
If the vertices A, B and C of ∆ABC have position vectors (1, 2, 3), (−1, 0, 0) and (0, 1, 2), respectively, what is the magnitude of ∠ABC?
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.
Find the vector from the origin O to the centroid of the triangle whose vertices are (1, −1, 2), (2, 1, 3) and (−1, 2, −1).
If \[\overrightarrow{AO} + \overrightarrow{OB} = \overrightarrow{BO} + \overrightarrow{OC} ,\] prove that A, B, C are collinear points.
Show that the vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k}\text{ and }- 4 \hat{i} + 6 \hat{j} - 8 \hat{k}\] are collinear.
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 \[\vec{a} \neq \vec{d} \text { and } \vec{b} \neq \vec{c}\] .
if `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2 hat"j" - 3hat"k" and hat"i" - 6hat"j" - hat"k"` respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether `vec"AB" and vec"CD"` are collinear or not.
If `vec"a"` and `vec"b"` are the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that BC = 1.5 BA.
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.
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:
Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2
