Advertisements
Advertisements
Question
Find the magnitude of two vectors `veca and vecb`, having the same magnitude and such that the angle between them is 60° and their scalar product is `1/2`.
Advertisements
Solution 1
We have, `theta = 60^circ, veca xx vecb = 1/2, |veca| = |vecb|`
If the angle between vectors `veca, vecb` is θ, then
Now, `costheta = (veca xx vecb)/(|veca||vecb|)`
`cos 60^circ = (1/2)/|veca|^2; 1/2 = 1/(2|veca|^2)`
`|veca|^2 = 1; |veca| = 1`
`|veca| = 1, |vecb| = 1`
Solution 2
Magnitude of two vectors `veca and vecb` is same
`|veca| = |vecb|`
`veca.vecb = |veca| |vecb| costheta`, θ is the angle between `veca and vecb`
Given `theta = 60^\circ and veca.vecb = 1/2`
`veca.vecb = |veca| |vecb| costheta`
`veca.vecb = |veca| |veca| cos60^\circ`
`1/2 = |veca|^2 xx 1/2`
`|veca|^2 = 1`
`|veca| = +- 1`
Since magnitude of a vector is not negative
so `|veca| = 1`
`|veca| = |vecb| = 1`
APPEARS IN
RELATED QUESTIONS
Find a vector `veca` of magnitude `5sqrt2` , making an angle of `π/4` with x-axis, `π/2` with y-axis and an acute angle θ with z-axis.
If `veca` is a nonzero vector of magnitude 'a' and λ a nonzero scalar, then λ`veca` is unit vector if ______.
If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, show that the vector `veca + vecb+ vecc` is equally inclined to `veca, vecb` and `vecc`.
If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, find the angle which `veca + vecb + vecc`make with `veca or vecb or vecc`
If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is `sqrt(3)`.
Find a vector of magnitude of 5 units parallel to the resultant of the vectors \[\vec{a} = 2 \hat{i} + 3 \hat{j} - \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} +\widehat{k} .\]
Find a vector \[\vec{r}\] of magnitude \[3\sqrt{2}\] units which makes an angle of \[\frac{\pi}{4}\] and \[\frac{\pi}{4}\] with y and z-axes respectively.
Write a vector of magnitude 12 units which makes 45° angle with X-axis, 60° angle with Y-axis and an obtuse angle with Z-axis.
Write the length (magnitude) of a vector whose projections on the coordinate axes are 12, 3 and 4 units.
Find a vector in the direction of \[\overrightarrow{a} = 2 \hat{i} - \hat{j} + 2 \hat{k} ,\] which has magnitude of 6 units.
Write two different vectors having same magnitude.
Write a vector in the direction of vector \[5 \hat{i} - \hat{j} + 2 \hat{k}\] which has magnitude of 8 unit.
Find a vector \[\overrightarrow{a}\] of magnitude \[5\sqrt{2}\], making an angle of \[\frac{\pi}{4}\] with x-axis, \[\frac{\pi}{2}\] with y-axis and an acute angle θ with z-axis.
If in a ∆ABC, A = (0, 0), B = (3, 3 \[\sqrt{3}\]), C = (−3\[\sqrt{3}\], 3), then the vector of magnitude 2 \[\sqrt{2}\] units directed along AO, where O is the circumcentre of ∆ABC is
Find all vectors of magnitude `10sqrt(3)` that are perpendicular to the plane of `hat"i" + 2hat"j" + hat"k"` and `-hat"i" + 3hat"j" + 4hat"k"`
A vector `vec"r"` is inclined at equal angles to the three axes. If the magnitude of `vec"r"` is `2sqrt(3)` units, find `vec"r"`.
Find a vector of magnitude 6, which is perpendicular to both the vectors `2hat"i" - hat"j" + 2hat"k"` and `4hat"i" - hat"j" + 3hat"k"`.
The vector in the direction of the vector `hat"i" - 2hat"j" + 2hat"k"` that has magnitude 9 is ______.
Let `vecalpha = hati + 2hatj - hatk, vecbeta = 2hati - hatj + 3hatk, vecγ = 2hati + hatj + 6hatk`. If `vecalpha` and `vecbeta` are both perpendicular to a vector `vecδ` and `vecδ. vecγ` = 10, then the magnitude of `vecδ` is
If the sum of two-unit vectors is a unit vector, then the magnitude of their difference is
Two equal forces acting at a point with an angle of 60° between them, if the resultant is equal `30sqrt(3)N`, the magnitude of the force will be
The area under a velocity-time curve represents the change in ______?
Which of the following statements is false about forces/ couple?
In a triangle ABC three forces of magnitudes `3vec(AB), 2vec(AC)` and `6vec(CB)` are acting along the sides AB, AC and CB respectively. If the resultant meets AC at D, then the ratio DC : AD will be equal to :
The magnitude of the vector `6hati - 2hatj + 3hatk` is ______.
Read the following passage and answer the questions given below:
|
Teams A, B, C went for playing a tug of war game. Teams A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area. Team A pulls with force F1 = `6hati + 0hatj kN`, Team B pulls with force F2 = `-4hati + 4hatj kN`, Team C pulls with force F3 = `-3hati - 3hatj kN`,
|
- What is the magnitude of the force of Team A ?
- Which team will win the game?
- Find the magnitude of the resultant force exerted by the teams.
OR
In what direction is the ring getting pulled?
Find a vector of magnitude 20 units parallel to the vector `2hati + 5hatj + 4hatk`.
