Advertisements
Advertisements
प्रश्न
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"`.
उत्तर
Let the given vectors
` overline(a)= 2hat"i" + 5hat"j" -4 hat"k", overline(b) = 5hat"i" +7hat"j"+5 hat "k" ,overline(c) = 4hat"i" +5hat"j" - 2 hat"k"`.
represent the co-terminous edges of the parallelopied
Consider `overline("a") . (bar "b" xx bar"c") = abs[(2 ,5,-4),(5 , 7, 5),(4 , 5, -2)]`
= 2 (-14-25) -5 (-10-20) -4 (25 - 28)
= - 78 +150 + 12 = 84
∴Volume of the parallelopiped = `abs(bar "a" .(bar"b" xx bar"c"))`
= 84 cubic units.
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 `
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.
Find the volume of the parallelopiped whose coterminus edges are given by vectors
`2hati+3hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`
If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`
(A) 100
(B) 101
(C) 110
(D) 109
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).
if `bara = 3hati - 2hatj+7hatk`, `barb = 5hati + hatj -2hatk`and `barc = hati + hatj - hatk` then find `bara.(barbxxbarc)`
Find the volume of a parallelopiped whose edges are represented by the vectors:
`vec a = 2 hat i - 3 hat j - 4 hat k`, `vec b = hat i + 2 hat j - hat k` and `vec c = 3 hat i + hat j + 2 hatk`
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 \[\left[ \vec{a} \vec{b} \vec{c} \right]\] , when \[\vec{a} = 2 \hat{i} - 3 \hat{j} , \vec{b} = \hat{i} + \hat{j} - \hat{k} \text{ and } \vec{c} = 3 \hat{i} - \hat{k}\]
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} , \vec{b} =\hat{ i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} + 2 \hat{k}\]
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} - 2 \hat{k}\]
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 11 \hat{i} , \vec{b} = 2 \hat{j} , \vec{c} = 13 \hat{k}\]
Show of the following triad of vector is coplanar:
\[\vec{a} = \hat {i} + 2 \hat{j} - \hat {k} , \vec{b} = 3 \hat {i} + 2 \hat{j} + 7 \hat {k} , \vec{c} = 5 \hat {i} + 6 \hat { j} + 5 \hat {k}\]
Show of the following triad of vector is coplanar:
\[\hat{a} = \hat{i} - 2 \hat {j} + 3 \hat {k} , \hat {b} = - 2 \hat {i} + 3 \hat {j} - 4 \hat { k}, \hat {c} = \hat { i} - 3 \hat { j} + 5 \hat { k }\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = \hat{i} + 2\hat { j} - 3 \hat {k} , \vec{b} = 3 \hat{i} + \lambda \hat {j} + \hat {k} , \vec{c} = \hat {i} + 2 \hat {j} + 2 \hat {k}\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = \hat {i} + 3 \hat {j} , \vec{b} = 5 \hat {k} , \vec{c} = \lambda \hat {i} - \hat {j}\]
Find the value of λ for which the four points with position vectors
\[-\hat { j} - \hat {k} , 4 \hat {i} + 5 \hat {j} + \lambda \hat {k} , 3 \hat {i} + 9 \hat {j} + 4 \hat {k} \text { and } - 4 \hat {i} + 4 \hat {j} + 4 \hat{k}\]
Prove that: \[\left( \vec{a} - \vec{b} \right) \cdot \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\} = 0\]
\[\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{and} \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},\]
If c1 = 1 and c2 = 2, find c3 which makes \[\vec{a,} \vec{b} \text { and } \vec{c}\] coplanar.
Find λ for which the points A (3, 2, 1), B (4, λ, 5), C (4, 2, −2) and D (6, 5, −1) are coplanar.
Write the value of \[\left[ 2 \hat { i } \ 3 \hat { j }\ 4 \hat { k } \right] .\]
Write the value of \[\left[ \hat {i} + \hat {j} \ \hat {j} + \hat {k} \ \hat {k} + \hat {i} \right] .\]
Find the values of 'a' for which the vectors
\[\vec{\alpha} = \hat {i} + 2 \hat {j} + \hat {k} , \vec{\beta} = a \hat {i} + \hat {j} + 2 \hat {k} \text { and } \vec{\gamma} = \hat {i} + 2 \hat {j} + a \hat { k }\] are coplanar.
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 \[\vec{c} = 3 \hat { i} + \hat {j} + 2 \hat {k}\].
If \[\vec{a}\] lies in the plane of vectors \[\vec{b} \text { and } \vec{c}\], then which of the following is correct?
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
Let \[\vec{a} = a_1 \hat { i }+ a_2 \hat {j} + a_3 \hat {k} , \vec{b} = b_1 \hat {i} + b_2 \hat { j } + b_3 \hat { k} \text { and } \vec{c} = c_1 \hat { i } + c_2 \hat{j } + c_3\text { k }\] be three non-zero vectors such that \[\vec{c}\] is a unit vector perpendicular to both \[\vec{a} \text { and } \vec{b}\]. If the angle between \[\vec{a} \text { and } \vec{b}\] is \[\frac{\pi}{6},\] , then
\[\begin{vmatrix}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{vmatrix}^2\] is equal to
If \[\left[ 2 \vec{a} + 4 \vec{b} \vec{c} \vec{d} \right] = \lambda\left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right],\] then λ + μ =
\[\left[ \vec{a} \vec{b} \vec{a} \times \vec{b} \right] + \left( \vec{a} . \vec{b} \right)^2 =\]
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 =
For non-zero vectors \[\vec{a,} \vec{b} \text { and }\vec{c}\] the relation \[\left| \left( \vec{a} \times \vec{b} \right) \cdot \vec{c} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \left| \vec{c} \right|\] holds good, if
\[\left( \vec{a} + 2 \vec{b} - \vec{c} \right) \cdot \left\{ \left( \vec{a} - \vec{b} \right) \times \left( \vec{a} - \vec{b} - \vec{c} \right) \right\}\] is equal to
Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:
`bar"a" = - 9hat"i" + 6hat"j" + 15hat"k"` , `bar"b" = 6hat"i" - 4hat"j" - 10hat"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"`
Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:
`bar"a" = 4hat"i" - hat"j" + 6hat"k"` , `bar"b" = 5hat"i" - 2hat"j" + 4hat"k"`
Prove by vector method, that the angle subtended on semicircle is a right angle.
If a line has the direction ratios 4, −12, 18, then find its direction cosines
Find the angle between the lines whose direction cosines l, m, n satisfy the equations 5l + m + 3n = 0 and 5mn − 2nl + 6lm = 0.
Find `bar"a".(bar"b" xx bar"c")` if `bar"a" = 3hat"i" - hat"j" + 4hat"k" , bar"b" = 2hat"i" + 3hat"j" - hat"k"` and `bar"c" = - 5hat"i" + 2hat"j" + 3hat"k"`
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.
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")`
The volume of the parallelepiped whose coterminus edges are `7hat"i" + lambdahat"j" - 3hat"k", hat"i" + 2hat"j" - hat"k", -3hat"i" + 7hat"j" + 5hat"k"` is 90 cubic units. Find the value of λ
If `vec"a", vec"b", vec"c"` are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume 4 cubic units, find the value of `(vec"a" + vec"b") * (vec"b" xx vec"c") + (vec"b" + vec"c")* (vec"c" xx vec"a") + (vec"c" + vec"a") * (vec"a" xx vec"b")`
If `vec"a" = hat"i" - hat"k", vec"b" = xhat"i" + hat"j" + (1 - x)hat"k", vec"c" = yhat"i" + xhat"j" + (1 + x - y)hat"k"`, show that `[(vec"a", vec"b", vec"c")]` depends on neither x nor y
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`
The volume of tetrahedron whose vertices are A(3, 7, 4), B(5, -2, 3), C(-4, 5, 6), D(1, 2, 3) is ______.
If the volume of the tetrahedron formed by the coterminous edges `bar"a", bar"b" and bar"c"` is 5, then the volume of the parallelopiped formed by the coterminous edges `bar"a" xx bar"b", bar"b" xx bar"c" and bar"c" xx bar"a"` is
If the scalar triple product of the vectors `-3hat"i" + 7hat"j" - 3hat"k", 3hat"i" - 7hat"j" + lambdahat"k" and 7hat"i" - 5hat"j" - 5hat"j"` is 272 then λ = ______.
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 ______.
If the points having the position vectors `2hat"i" + hat"j" - hat"k", -hat"j", 4hat"i" + 4hat"k"` and `lambdahat"i" + hat"k"` lie on the same plane, then λ is equal to ______.
Let `bar"a", bar"b", bar"c"` be three vectors such that `bar"a" ≠ 0`, and `bar"a" xx bar"b" = 2bar"a" xx bar"c", |bar"a"| = |bar"c"| = 1, |bar"b"| = 4` and `|bar"b" xx bar"c"| = sqrt(15)`. If `bar"b" - 2bar"c" = lambdabar"a"`, then λ is equal to ______.
If θ is the angle between the unit vectors `bar"a"` and `bar"b"`, the `cos theta = theta/2` = ______.
If the direction cosines of a line are `(1/c, 1/c, 1/c)` then ______.
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 `bara` and `barb` are orthogonal, parallel or neither.
`bara = - 3/5 hati + 1/2 hatj + 1/3 hatk, barb = 5hati + 4hatj + 3hatk`
Determine whether `bara and barb` are orthogonal, parallel or neither.
`bara = -3/5hati + 1/2hatj + 1/3hatk, barb = 5hati + 4hatj + 3hatk`
If `2hati + 3hatj, hati + hatj + hatk` and `λhati + 4hatj + 2hatk` taken in order are coterminous edges of a parallelopiped of volume 2 cu. units, then find the value of λ.
Find the volume of the parallelopiped whose coterminous edges are `2hati - 3hatj, hati + hatj - hatk` and `3hati - hatk`.
If `bar"u" = hat"i" - 2hat"j" + hat"k" , bar"v" = 3hat"i" + hat"k"` and `bar"w" = hat"j" - hat"k"` are given vectors, then find `[bar"u" xx bar"v" bar"u" xx bar"w" bar"v" xx bar"w"]`
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`
Determine whether `\bb(bara and barb)` are orthogonal, parallel or neither.
`bara = -3/5 hati + 1/2 hatj + 1/3 hatk, barb = 5hati + 4hatj + 3hatk `
