Advertisements
Advertisements
प्रश्न
`hati "and" hatj` are unit vectors along x- and y-axis respectively. What is the magnitude and direction of the vectors `hati+hatj` and `hati-hatj` ? What are the components of a vector `A = 2hati + 3hatj` along the directions of `hati + hatj` and `hati - hatj` ? [You may use graphical method]
Advertisements
उत्तर
`hati+hatj = sqrt((1)^2 + (1)^2+2xx1xx1xxcos 90^@) = sqrt2` = 1.414 unit
`tan theta = 1/1 = 1 :. theta = 45^@`
So the vector `hati+hatj` makes an angle `45^@` with x-axis
`|hati-hatj| = sqrt((1)^2 +(2)^2 - 2 xx 1xx1xxcos 90^@`
`= sqrt2` = 1.414 units
The vector `hati -hatj` makess an angle `-45^@` with x-axis
Let us now determined the component of `vecA = 2hati+3hatj` in the direction of `hati+hatj`
Let `vecB = hati+hatj`
`vecA.vecB = AB cos theta = (Acostheta)B`
So the component of `vecA` in the direction of `vecB` = `(vecA.vecB)/B`
`=((2hati+3hatj).(hati+hatj))/sqrt((1)^2+(1)^2) = (2hati.hati+2hati.hatj+3hatj.hati+3hatj.hatj)/sqrt2 = 5/sqrt2 units`
Component of `vecA` in the drection of `hati-hatj = ((2hati+3hatj).(hati-hatj))/sqrt2 = -1/sqrt2` units
APPEARS IN
संबंधित प्रश्न
Answer the following question.
Show that `vec"a" = (hat"i" - hat"j")/sqrt2` is a unit vector.
Answer the following question.
If `vec"v"_1 = 3hat"i" + 4hat"j" + hat"k" and vec"v"_2 = hat"i" - hat"j" - hat"k"`, determine the magnitude of `vec"v"_1 + vec"v"_2`.
For `vec v_1 = 2 hat i - 3 hat j` and `vec v_2 = -6hat i + 5 hat j`, determine the magnitude and direction of `vec v_1 + vec v_2`.
Find a vector which is parallel to `vec"v" = hat"i" - 2hat"j"` and has a magnitude 10.
Show that vectors `vec a = 2 hat i + 5 hat j - 6 hat k` and `vec b = hat i + 5/2 hat j - 3 hat k` are parallel.
Figure shows the orientation of two vectors u and v in the XY plane.
If `u = ahati + bhatj` and `v = phati + qhatj`
which of the following is correct?
What is the process of splitting a single vector into two or more vectors along specific directions called?
What is the defining feature of rectangular components of a vector?
A vector \[\vec V\] = 3î + 4ĵ. What is the magnitude of this vector?
Which of the following correctly represents a vector in a 3D coordinate system using unit vectors?
What is the result of dividing a non-zero vector \[\vec V\] by its own magnitude ∣\[\vec V\]∣?
The component of a vector may be ______.
The vector projection of a vector 3\[\hat i\] + 4\[\hat k\] on Y-axis is ______.
A vector is represented by 3\[\hat i\] + \[\hat j\] + 2\[\hat k\]. Its length in XY plane is ______.
A particle is simultaneously acted by two forces equal to 4 N and 3 N. The net force on the particle is ______.
