Advertisements
Advertisements
Question
Find a vector whose length is 3 and which is perpendicular to the vector \[\vec{a} = 3 \hat{ i } + \hat{ j } - 4 \hat{ k } \text{ and } \vec{b} = 6 \hat{ i } + 5 \hat{ j } - 2 \hat{ k } .\]
Advertisements
Solution
\[\text{ Given } : \]
\[ \vec{a} = 3 \hat{ i } + \hat{ j } - 4 \hat{ k } \]
\[ \vec{b} = 6 \hat{ i } + 5 \hat{ j } - 2 \hat{ k } \]
\[ \therefore a^\to \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat { k } \\ 3 & 1 & - 4 \\ 6 & 5 & - 2\end{vmatrix}\]
\[ = \left( - 2 + 20 \right) \hat{ i } - \left( - 6 + 24 \right) \hat{ j } + \left( 15 - 6 \right) \hat{ k } \]
\[ = 18 \hat{ i } - 18 \hat{ j } + 9 \hat{ k } \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{{18}^2 + \left( - {18}^2 \right) + 9^2}\]
\[ = \sqrt{729}\]
\[ = 27\]
\[\text{ Required vector } = 3 \times \left\{ \frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|} \right\}\]
\[ = 3 \times \frac{18 \hat{ i } - 18 \hat{ j } + 9 \hat{ k } }{27}\]
\[ = \frac{3\left( 2 \hat{ i } - 2 \hat{ j } + \hat{ k } \right)}{3}\]
\[ = 2 \hat{ i } - 2 \hat{ j } + \hat{ k } \]
APPEARS IN
RELATED QUESTIONS
If `veca = 2hati + 2hatj + 3hatk, vecb = -veci + 2hatj + hatk and vecc = 3hati + hatj` are such that `veca + lambdavecb` is perpendicular to `vecc`, then find the value of λ.
If a unit vector `veca` makes an angles `pi/3` with `hati, pi/4` with `hatj` and an acute angle θ with `hatk`, then find θ and, hence the compounds of `veca`.
Show that `(veca - vecb) xx (veca + vecb) = 2(veca xx vecb)`.
Find λ and μ if `(2hati + 6hatj + 27hatk) xx (hati + lambdahatj + muhatk) = vec0`.
If either `veca = vec0` or `vecb = vec0`, then `veca xxvecb = vec0`. Is the converse true? Justify your answer with an example.
Find the area of the parallelogram whose adjacent sides are determined by the vector `veca = hati - hatj + 3hatk` and `vecb = 2hati - 7hatj + hatk`.
If θ is the angle between two vectors `hati - 2hatj + 3hatk and 3hati - 2hatj + hatk` find `sin theta`
Let `veca = 4hati + 5hatj - hatk`, `vecb = hati - 4hatj + 5hatk` and `vecc = 3hati + hatj - hatk`. Find a vector `vecd` which is perpendicular to both `vecc` and `vecb and vecd.veca = 21`
If \[\vec{a} = 3 \hat { i } + 4 \hat { j } \text{ and } \vec{b} = \hat { i } + \hat{ j } + \hat{ k } ,\] find the value of \[\left| \vec{a} \times \vec{b} \right| .\]
Find the magnitude of \[\vec{a} = \left( 3 \hat{ k } + 4 \hat{ j } \right) \times \left( \hat{ i } + \hat{ j } - \hat{ k } \right) .\]
Find the area of the parallelogram determined by the vector \[2 \hat{ i } + \hat{ j } + 3 \hat{ k } \text{ and } \hat{ i } - \hat{ j } \] .
Find the area of the parallelogram whose diagonals are \[2 \hat{ i } + 3 \hat{ j } + 6 \hat{ k } \text{ and } 3 \hat{ i } - 6 \hat{ j } + 2 \hat{ k } \]
If \[\vec{a} = 2 \hat{ i } + 5 \hat{ j } - 7 \hat{ k } , \vec{b} = - 3 \hat{ i } + 4 \hat{ j } + \hat{ k } \text{ and } \vec{c} = \hat{ i } - 2 \hat{ j } - 3 \hat{ k } ,\] compute \[\left( \vec{a} \times \vec{b} \right) \times \vec{c} \text{ and } \vec{a} \times \left( \vec{b} \times \vec{c} \right)\] and verify that these are not equal.
Given \[\vec{a} = \frac{1}{7}\left( 2 \hat{ i } + 3 \hat{ j } + 6 \hat{ k } \right), \vec{b} = \frac{1}{7}\left( 3 \hat{ i } - 6 \hat{ j } + 2 \hat{ k } \right), \vec{c} = \frac{1}{7}\left( 6 \hat{ i } + 2 \hat{ j } - 3 \hat{ k }\right), \hat{ i } , \hat{ j } , \hat{ k } \] being a right handed orthogonal system of unit vectors in space, show that \[\vec{a} , \vec{b} , \vec{c}\] is also another system.
if \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 7 \text{ and } \vec{a} \times \vec{b} = 3 \hat{ i } + 2 \hat{ j } + 6 \hat{ k } ,\] find the angle between \[\vec{a} \text{ and } \vec{b} .\]
If a, b, c are the lengths of sides, BC, CA and AB of a triangle ABC, prove that \[\vec{BC} + \vec{CA} + \vec{AB} = \vec{0}\] and deduce that \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .\]
if \[\vec{a} = \hat{ i }- 2\hat{ j } + 3 \hat{ k } , \text{ and } \vec{b} = 2 \hat{ i } + 3 \hat{ j } - 5 \hat{ k } ,\] then find \[\vec{a} \times \vec{b} .\] Verify th at \[\vec{a} \text{ and } \vec{a} \times \vec{b}\] are perpendicular to each other.
For any two vectors \[\vec{a} \text{ and } \vec{b}\] , prove that \[\left| \vec{a} \times \vec{b} \right|^2 = \begin{vmatrix}\vec{a} . \vec{a} & & \vec{a} . \vec{b} \\ \vec{b} . \vec{a} & & \vec{b} . \vec{b}\end{vmatrix}\]
Find the area of the triangle formed by O, A, B when \[\vec{OA} = \hat{ i } + 2 \hat{ j } + 3 \hat{ k } , \vec{OB} = - 3 \hat{ i } - 2 \hat{ j }+ \hat{ k } .\]
Find all vectors of magnitude \[10\sqrt{3}\] that are perpendicular to the plane of \[\hat{ i } + 2 \hat{ j } + \hat{ k } \] and \[- \hat { i } + 3 \hat{ j } + 4 \hat{ k } \] .
Write the value \[\left( \hat{ i } \times \hat{ j } \right) \cdot \hat{ k } + \hat{ i } \cdot \hat{ j } .\]
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3}\text{ and } \vec{a} . \vec{b} = 1,\] find the angle between.
Write the value of \[\hat{ i } \times \left(\hat{ j } \times \hat{ k } \right) .\]
If \[\vec{a}\] is a unit vector such that \[\vec{a} \times \hat{ i } = \hat{ j } , \text{ find } \vec{a} . \hat{ i } \] .
Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes 1 and 2 respectively and when \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3} .\]
Find a vector of magnitude \[\sqrt{171}\] which is perpendicular to both of the vectors \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] and \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] .
If \[\vec{a}\] is any vector, then \[\left( \vec{a} \times \hat{ i } \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times \hat{ k } \right)^2 =\]
A unit vector perpendicular to both \[\hat{ i } + \hat{ j } \text{ and } \hat{ j } + \hat{ k } \] is
If θ is the angle between the vectors \[2 \hat{ i } - 2 \hat{ j} + 4 \hat{ k } \text{ and } 3 \hat{ i } + \hat { j } + 2 \hat{ k } ,\] then sin θ =
Find a unit vector perpendicular to both the vectors `veca and vecb` , where `veca = hat i - 7 hatj +7hatk` and `vecb = 3hati - 2hatj + 2hatk` .
Let `veca = hati + hatj, vecb = hati - hatj` and `vecc = hati + hatj + hatk`. If `hatn` is a unit vector such that `veca.hatn` = 0 and `vecb.hatn` = 0, then find `|vecc.hatn|`.
Let `veca = 2hati + hatj - 2hatk, vecb = hati + hatj`. If `vecc` is a vector such that `veca . vecc = \|vecc|, |vecc - veca| = 2sqrt(2)` and the angle between `veca xx vecb` and `vecc` is 30°, then `|(veca xx vecb) xx vecc|` equals ______.
