Advertisements
Advertisements
Question
Using vectors find the area of the triangle with vertices, A (2, 3, 5), B (3, 5, 8) and C (2, 7, 8).
Advertisements
Solution
\[\text { Let } \vec{a} , \vec{b} \text{ and } \vec{c}^{} \text{ be the position vectors of A, B and C, respectively . Then, } \]
\[ \vec{a} = 2 \hat{ i } + 3 \hat{ j } + 5 \hat{ k } \]
\[ \vec{b} = 3 \hat{ i } + 5\hat{ j } + 8 \hat{ k } \]
\[ \vec{c} = 2 \hat{ i } + 7 \hat{ j } + 8 \hat{ k } \]
\[\text{ Now } , \]
\[ \vec{AB} = \vec{b} - \vec{a} \]
\[ = \hat{ i } + 2 \hat{ j } + 3 \hat{ k } \]
\[ \vec{AC} = \vec{c} - \vec{a} \]
\[ = 0 \hat{ i }+ 4 \hat{ j } + 3 \hat{ k } \]
\[ \therefore \vec{AB} \times \vec{AC} = \begin{vmatrix}\hat{ i }& \hat{ j } & \hat{ k } \\ 1 & 2 & 3 \\ 0 & 4 & 3\end{vmatrix}\]
\[ = - 6 \hat{ i } - 3 \hat{ j } + 4 \hat{ k } \]
\[ \Rightarrow \left| \vec{AB} \times \vec{AC} \right| = \sqrt{36 + 9 + 16}\]
\[ = \sqrt{61}\]
\[\text{ Area of triangleABC } =\frac{1}{2}\left| \vec{AB} \times \vec{AC} \right|\]
\[ =\frac{\sqrt{61}}{2} \text{ sq. units } \]
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 θ is the angle between two vectors `hati - 2hatj + 3hatk and 3hati - 2hatj + hatk` find `sin theta`
If \[\vec{a} = 3 \hat { i } + 4 \hat { j } \text{ and } \vec{b} = \hat { i } + \hat{ j } + \hat{ k } ,\] find the value of \[\left| \vec{a} \times \vec{b} \right| .\]
If \[\vec{a} = 2 \hat{ i } + \hat{ k } , \vec{b} = \hat { i } + \hat{ j } + \hat{ k } ,\] find the magnitude of \[\vec{a} \times \vec{b} .\]
Find the magnitude of \[\vec{a} = \left( 3 \hat{ k } + 4 \hat{ j } \right) \times \left( \hat{ i } + \hat{ j } - \hat{ k } \right) .\]
Find a vector whose length is 3 and which is perpendicular to the vector \[\vec{a} = 3 \hat{ i } + \hat{ j } - 4 \hat{ k } \text{ and } \vec{b} = 6 \hat{ i } + 5 \hat{ j } - 2 \hat{ k } .\]
Find the area of the parallelogram whose diagonals are \[2 \hat{ i }+ \hat{ k } \text{ and } \hat{ i } + \hat{ j } + \hat{ k } \]
Define \[\vec{a} \times \vec{b}\] and prove that \[\left| \vec{a} \times \vec{b} \right| = \left( \vec{a} . \vec{b} \right)\] tan θ, where θ is the angle between \[\vec{a} \text{ and } \vec{b}\] .
The two adjacent sides of a parallelogram are \[2 \hat{ i } - 4 \hat{ j } + 5 \hat{ k } \text{ and } \hat{ i } - 2 \hat{ j } - 3\hat{ k } .\]\ Find the unit vector parallel to one of its diagonals. Also, find its area.
Define vector product of two vectors.
Write the value of \[\hat{ i } × \left( \hat{ j } + \hat{ k } \right) + \hat{ j } × \left( \hat{ k } + \hat{ i } \right) + \hat{ k } × \left( \hat{ i } + \hat{ j } \right) .\]
Write a unit vector perpendicular to \[\hat{ i } + \hat{ j } \text{ and } \hat{ j } + \hat{ k } .\]
If \[\vec{r} = x \hat{ i } + y \hat{ j } + z \hat{ k } ,\] then write the value of \[\left| \vec{r} \times \hat{ i } \right|^2 .\]
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors such that \[\vec{a} \times \vec{b}\] is also a unit vector, find the angle between \[\vec{a} \text{ and } \vec{b}\] .
If \[\vec{a} \text{ and } \vec{b}\] are two vectors such that \[\left| \vec{a} . \vec{b} \right| = \left| \vec{a} \times \vec{b} \right|,\] write the angle between \[\vec{a} \text{ and } \vec{b} .\]
If \[\vec{a}\] is a unit vector such that \[\vec{a} \times \hat{ i } = \hat{ j } , \text{ find } \vec{a} . \hat{ i } \] .
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 the area of the parallelogram determined by the vectors \[2 \hat{ i } \text{ and } 3 \hat{ j } .\]
Write the number of vectors of unit length perpendicular to both the vectors \[\vec{a} = 2 \hat{ i } + \hat{ j } + 2 \hat{ k } \text{ and } \vec{b} = \hat{ j } + \hat{ k } \] .
Write the angle between the vectors \[\vec{a} \times \vec{b}\] and \[\vec{b} \times \vec{a}\] .
If \[\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}\] and \[\vec{a} \times \vec{b} = \vec{a} \times \vec{c,} \vec{a} \neq 0,\] then
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 \[\vec{\alpha} =\]
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
If \[\vec{a} = 2 \hat{ i } - 3 \hat{ j } - \hat{ k } \text{ and } \vec{b} = \hat{ i } + 4 \hat{ j } - 2 \hat{ k
} , \text{ then } \vec{a} \times \vec{b}\] is
If \[\hat{ i } , \hat{ j } , \hat{ k } \] are unit vectors, then
If \[\left| \vec{a} \times \vec{b} \right| = 4, \left| \vec{a} \cdot \vec{b} \right| = 2, \text{ then } \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 =\]
The value of \[\hat{ i } \cdot \left( \hat{ j } \times \hat{ k } \right) + \hat{ j } \cdot \left( \hat{ i } \times \hat{ k } \right) + \hat{ k } \cdot \left( \hat{ i } \times \hat{ j } \right),\] is
(a) If `veca = hati - 2j + 3veck , vecb = 2hati + 3hatj - 5hatk,` prove that `veca and vecaxxvecb` are perpendicular.
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 number of vectors of unit length perpendicular to the vectors `vec"a" = 2hat"i" + hat"j" + 2hat"k"` and `vec"b" = hat"j" + hat"k"` is ______.
Let `veca` and `vecb` be two unit vectors and θ is the angle between them, Then `veca + vecb` is a unit vector if-
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.
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`.
If `veca = hati + hatj + hatk` and `vecb = hati + 2hatj + 3hatk` then find a unit vector perpendicular to both `veca + vecb` and `veca - vecb`.
If `veca` and `vecb` are two non-zero vectors such that `|veca xx vecb| = veca.vecb`, find the angle between `veca` and `vecb`.
