Advertisements
Advertisements
Question
Write the expression for the area of the parallelogram having \[\vec{a} \text{ and } \vec{b}\] as its diagonals.
Advertisements
Solution
\[\text{ Given:} \vec{a} \text{ and } \vec{b} \text{ are diagonals of a parallelogram } .\]
\[\text{ Area of the parallelogram } =\frac{1}{2}\left| \vec{a} \times \vec{b} \right|\]
APPEARS IN
RELATED QUESTIONS
Find a unit vector perpendicular to each of the vector `veca + vecb` and `veca - vecb`, where `veca = 3hati + 2hatj + 2hatk` and `vecb = hati + 2hatj - 2hatk`.
Find λ and μ if `(2hati + 6hatj + 27hatk) xx (hati + lambdahatj + muhatk) = vec0`.
Let the vectors `veca, vecb, vecc` given as `a_1hati + a_2hatj + a_3hatk, b_1hati + b_2hatj + b_3hatk, c_1hati + c_2hatj + c_3hatk` Then show that = `veca xx (vecb+ vecc) = veca xx vecb + veca xx vecc.`
If θ is the angle between two vectors `hati - 2hatj + 3hatk and 3hati - 2hatj + hatk` find `sin theta`
If A, B, C are three non- collinear points with position vectors `vec a, vec b, vec c`, respectively, then show that the length of the perpendicular from Con AB is `|(vec a xx vec b)+(vec b xx vec c) + (vec b xx vec a)|/|(vec b - vec a)|`
Find a unit vector perpendicular to both the vectors \[\vec{a} + \vec{b} \text { and } \vec{a} - \vec{b}\] ,where \[\vec{a} = \hat{i}+ \hat{j} + \hat{k} , \vec{b} =\hat {i} + 2 \hat{j} + 3 \hat{k}\].
\[\text{ If } \vec{a} = \hat { i } + 3 \hat { j } - 2 \hat { k } \text{ and } \vec{b} = - \hat { i } + 3 \hat { k } , \text{ find } \left| \vec{a} \times \vec{b} \right| .\]
Find the area of the parallelogram determined by the vector \[2 \hat{ i } \text{ and } 3 \hat{ j } \] .
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 \[\vec{a} \times \vec{b} = \vec{b} \times \vec{c} \neq 0,\] then show that \[\vec{a} + \vec{c} = m \vec{b} ,\] where m is any scalar.
If \[\vec{p} \text{ and } \vec{q}\] are unit vectors forming an angle of 30°; find the area of the parallelogram having \[\vec{a} = \vec{p} + 2 \vec{q} \text{ and } \vec{b} = 2 \vec{p} + \vec{q}\] as its diagonals.
Find the area of the triangle formed by O, A, B when \[\vec{OA} = \hat{ i } + 2 \hat{ j } + 3 \hat{ k } , \vec{OB} = - 3 \hat{ i } - 2 \hat{ j }+ \hat{ k } .\]
If either \[\vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0} , \text{ then } \vec{a} \times \vec{b} = \vec{0} .\] Is the converse true? Justify your answer with an example.
Using vectors, find the area of the triangle with vertice A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) .
If \[\vec{a}\] is a unit vector such that \[\vec{a} \times \hat{ i } = \hat{ j } , \text{ find } \vec{a} . \hat{ i } \] .
If \[\vec{c}\] is a unit vector perpendicular to the vectors \[\vec{a} \text{ and } \vec{b} ,\] write another unit vector perpendicular to \[\vec{a} \text{ and } \vec{b} .\]
Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] with magnitudes 1 and 2 respectively and when \[\left| \vec{a} \times \vec{b} \right| = \sqrt{3} .\]
Vectors \[\vec{a} \text{ and } \vec{b}\] \[\left| \vec{a} \right| = \sqrt{3}, \left| \vec{b} \right| = \frac{2}{3}\text{ and } \left( \vec{a} \times \vec{b} \right)\] is a unit vector. Write the angle between \[\vec{a} \text{ and } \vec{b}\] .
Write the value of \[\left( \hat{ i } \times \hat{ j } \right) \cdot \hat{ k } + \left( \hat{ j } + \hat{ k } \right) \cdot \hat{ j } \]
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 \[\vec{a} = \hat{ i } + 2 \hat{ j } - 3 \hat{ k } \] .
Write the number of vectors of unit length perpendicular to both the vectors \[\vec{a} = 2 \hat{ i } + \hat{ j } + 2 \hat{ k } \text{ and } \vec{b} = \hat{ j } + \hat{ k } \] .
If \[\vec{a}\] is any vector, then \[\left( \vec{a} \times \hat{ i } \right)^2 + \left( \vec{a} \times \hat{ j } \right)^2 + \left( \vec{a} \times \hat{ k } \right)^2 =\]
The vector \[\vec{b} = 3 \hat { i }+ 4 \hat {k }\] is to be written as the sum of a vector \[\vec{\alpha}\] parallel to \[\vec{a} = \hat {i} + \hat {j}\] and a vector \[\vec{\beta}\] perpendicular to \[\vec{a}\]. Then \[\vec{\alpha} =\]
Vectors \[\vec{a} \text{ and } \vec{b}\] are inclined at angle θ = 120°. If \[\left| \vec{a} \right| = 1, \left| \vec{b} \right| = 2,\] then \[\left[ \left( \vec{a} + 3 \vec{b} \right) \times \left( 3 \vec{a} - \vec{b} \right) \right]^2\] is equal to
If \[\vec{a} = \hat{ i } + \hat{ j } - \hat{ k } , \vec{b} = - \hat{ i } + 2\hat{ j } + 2 \hat{ k } \text{ and } \vec{c} = - \hat{ i } + 2 \hat{ j } - \hat{ k } ,\] then a unit vector normal to the vectors \[\vec{a} + \vec{b} \text{ and } \vec{b} - \vec{c}\] is
Find a unit vector perpendicular to both the vectors `veca and vecb` , where `veca = hat i - 7 hatj +7hatk` and `vecb = 3hati - 2hatj + 2hatk` .
The two adjacent sides of a parallelogram are represented by vectors `2hati - 4hatj + 5hatk` and `hati - 2hatj - 3hatk`. Find the unit vector parallel to one of its diagonals, Also, find the area of the parallelogram.
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 ______.
If `|veca xx vecb| = sqrt(3)` and `veca.vecb` = – 3, then angle between `veca` and `vecb` is ______.
Find the area of a parallelogram whose adjacent sides are determined by the vectors `veca = hati - hatj + 3hatk` and `vecb = 2hati - 7hatj + hatk`.
Find the area of the parallelogram whose diagonals are `hati - 3hatj + hatk` and `hati + hatj + hatk`.
