Advertisements
Advertisements
प्रश्न
If \[\vec{a} = 2\hat{ i} - 3 \hat { j} + 5 \hat { k} , \vec{b} = 3 \hat {i} - 4 \hat {j} + 5 \hat {k} \text { and } \vec{c} = 5\hat { i } - 3 \hat {j}- 2 \hat{k},\] then the volume of the parallelopiped with conterminous edges \[\vec{a} + \vec{b,} \vec{b} + \vec{c,} \vec{c} + \vec{a}\] is
विकल्प
2
1
-1
0
None of these
Advertisements
उत्तर
We have
\[ \vec{a} + \vec{b} = \left( 2 \hat {i} - 3 \hat {j} + 5 \hat { k} \right) + \left( 3 \hat {i} - 4 \hat {j} + 5 \hat {k} \right) = 5 \hat {i} - 7 \hat {j} + 10 \hat {k} \]
\[ \vec{b} + \vec{c} = \left( 3 \hat {i} - 4 \hat {j} + 5 \hat {k} \right) + \left( 5 \hat {i} - 3 \hat {j} - 2 \hat {k} \right) = 8 \hat {i} - 7 \hat {j} + 3 \hat {k } \]
\[ \vec{c} + \vec{a} = \left( 5\hat { i } - 3 \hat {j} - 2 \hat {k} \right) + \left( 2 \hat {i} - 3 \hat {j} + 5 \hat {k} \right) = 7 \hat {i} - 6 \hat {j} + 3 \hat {k} \]
\[\text { We know that the volume of a parallelopiped whose three adjacent adges are } \vec{a} + \vec{b} , \vec{b} + \vec{c} \text { and } \vec{c} + \vec{a} \text { is equal to } \left| \left[ \vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a} \right] \right| . \]
We have
\[\left[ \vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a} \right] = \begin{vmatrix}5 & - 7 & 10 \\ 8 & - 7 & 3 \\ 7 & - 6 & 3\end{vmatrix}\]
\[ \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} = 5\left( - 21 + 18 \right) + 7\left( 24 - 21 \right) + 10\left( - 48 + 49 \right)\]
\[ = \left( 5 \times - 3 \right) + \left( 7 \times 3 \right) + \left( 10 \times 1 \right)\]
\[ = 16\]
\[ \therefore \text { Volume of parallelopiped } = \left| 16 \right| = 16\]
APPEARS IN
संबंधित प्रश्न
Prove that the volume of a parallelopiped with coterminal edges as ` bara ,bar b , barc `
Hence find the volume of the parallelopiped with coterminal edges `bar i+barj, barj+bark `
If `bar c = 3bara- 2bar b ` Prove that `[bar a bar b barc]=0`
Prove that, for any three vector `veca,vecb,vecc [vec a+vec b,vec b+vec c,vecc+veca]=2[veca vecb vecc]`
Show that the four points A(4, 5, 1), B(0, –1, –1), C(3, 9, 4) and D(–4, 4, 4) are coplanar.
Find λ, if the vectors `veca=hati+3hatj+hatk,vecb=2hati−hatj−hatk and vecc=λhatj+3hatk` are coplanar.
if `bara = 3hati - 2hatj+7hatk`, `barb = 5hati + hatj -2hatk`and `barc = hati + hatj - hatk` then find `bara.(barbxxbarc)`
Prove that a necessary and sufficient condition for three vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that \[l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\]
Evaluate the following:
\[\left[\hat{i}\hat{j}\hat{k} \right] + \left[ \hat{j}\hat{k}\hat {i} \right] + \left[ \hat{k}\hat{i} \hat{j} \right]\]
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 11 \hat{i} , \vec{b} = 2 \hat{j} , \vec{c} = 13 \hat{k}\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = \hat{i} - \hat{j} + \hat{k} , \vec{b} = 2 \hat {i} + \hat {j} - \hat {k} , \vec{c} = \lambda\hat { i} - \hat {j} + \lambda \hat {k}\]
Show that four points whose position vectors are
\[6 \hat { i} - 7 \hat {j} , 16 \hat { i} - 19 \hat { j} - 4 \hat {k} , 3 \hat {i} - 6 \hat {k} , 2 \hat { i} - 5 \hat {j}+ 10 \hat {k}\]
Prove that: \[\left( \vec{a} - \vec{b} \right) \cdot \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\} = 0\]
Write the value of \[\left[ \hat {i} - \hat {j} \hat {j} - \hat {k} \hat {k} - \hat {i} \right] .\]
Find the volume of the parallelopiped with its edges represented by the vectors \[\hat {i} + \hat {j} , \hat {i} + 2 \hat {j} \text { and } \hat {i} + \hat {j} + \pi k .\]
If \[\vec{a,} \vec{b}\] \[\text { are non-collinear vectors, then find the value of} \left[ \vec{a} \vec{b}\hat { i} \right] \hat{i} + \left[ \vec{a} \vec{b} \hat {j} \right] \hat {j} + \left[ \vec{a} \vec{b} \hat {k} \right] \hat {k} .\]
If \[\vec{a,} \vec{b,} \vec{c}\] are non-coplanar vectors, then find the value of \[\frac{\vec{a} \cdot \left( \vec{b} \times \vec{c} \right)}{\left( \vec{c} \times \vec{a} \right) \cdot \vec{b}} + \frac{\vec{b} \cdot \left( \vec{a} \times \vec{c} \right)}{\vec{c} \cdot \left( \vec{a} \times \vec{b} \right)}\].
Find \[\vec{a} . \left( \vec{b} \times \vec{c} \right)\], if \[\vec{a} = 2 \hat {i} + \hat {j} + 3 \hat {k} , \vec{b} = - \hat {i} + 2 \hat {j} + \hat {k}\] and \[\vec{c} = 3 \hat { i} + \hat {j} + 2 \hat {k}\].
The value of \[\left[ \vec{a} - \vec{b} , \vec{b} - \vec{c} , \vec{c} - \vec{a} \right], \text { where } \left| \vec{a} \right| = 1, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 3, \text { is }\]
If \[\vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = 0\] for some non-zero vector \[\vec{r} ,\] then the value of \[\left[ \vec{a} \vec{b} \vec{c} \right],\] is
For any three vectors \[\vec{a,} \vec{b,} \vec{c}\] the expression \[\left( \vec{a} - \vec{b} \right) . \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\}\] equals
If the vectors \[4 \hat { i} + 11 \hat {j} + m \hat {k} , 7 \hat { i} + 2 \hat { j} + 6 \hat {k} \text { and } \hat {i} + 5 \hat {j} + 4 \hat {k}\] are coplanar, then m =
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{b} + \vec{c} \right) \times \left( \vec{a} + \vec{b} + \vec{c} \right) =\]
Find the volume of the parallelopiped, if the coterminus edges are given by the vectors `2hat"i" + 5hat"j" -4 hat"k", 5hat"i" +7hat"j"+5 hat "k" , 4hat"i" +5hat"j" - 2 hat"k"`.
Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:
`bar"a" = 2hat"i" + 3hat"j" - hat"k"` , `bar"b" = 5hat"i" - 2hat"j" + 4hat"k"`
Find the angle between the lines whose direction cosines l, m, n satisfy the equations 5l + m + 3n = 0 and 5mn − 2nl + 6lm = 0.
If the vectors `3hat"i" + 5hat"k", 4hat"i" + 2hat"j" - 3hat"k"` and `3hat"i" + hat"j" + 4hat"k"` are the coterminus edges of the parallelopiped, then find the volume of the parallelopiped.
Using properties of scalar triple product, prove that `[(bar"a" + bar"b", bar"b" + bar"c", bar"c" + bar"a")] = 2[(bar"a", bar"b", bar"c")]`.
If `vec"a" = hat"i" - 2hat"j" + 3hat"k", vec"b" = 2hat"i" + hat"j" - 2hat"k", vec"c" = 3hat"i" + 2hat"j" + hat"k"`, find `vec"a" * (vec"b" xx vec"c")`
Let `vec"a", vec"b", vec"c"` be three non-zero vectors such that `vec"c"` is a unit vector perpendicular to both `vec"a"` and `vec"b"`. If the angle between `vec"a"` and `vec"b"` is `pi/6`, show that `[(vec"a", vec"b", vec"c")]^2 = 1/4|vec"a"|^2|vec"b"|^2`
If the volume of tetrahedron whose vertices are A(0, 1, 2), B(2, -3, 0), C(1, 0, 2) and D(-2,-3,lambda) is `7/3` cu.units, then the value of λ is ______.
Let v = `2hati + hatj - hatk` and w = `hati + 3hatk`. If u is a unit vector, then maximum value of scalar triple product [u v w] is ______.
Determine whether `bb(bara and barb)` are orthogonal, parallel or neither.
`bar a = -3/5hati + 1/2hatj + 1/3hatk, barb = 5hati + 4hatj + 3hatk`
If the points A(1, 2, 3), B(–1, 1, 2), C(2, 3, 4) and D(–1, x, 0) are coplanar find the value of x.
If `barc = 3bara - 2barb` and `[bara barb + barc bara + barb + barc]` = 0 then prove that `[bara barb barc]` = 0
If a vector has direction angles 45ºand 60º find the third direction angle.
Determine whether `bara and barb` is orthogonal, parallel or neither.
`bara = -3/5hati + 1/2hatj + 1/3hatk, barb = 5hati + 4hatj + 3hatk`
Find the volume of a tetrahedron whose vertices are A(−1, 2, 3) B(3, −2, 1), C (2, 1, 3) and D(−1, −2, 4).
