Advertisements
Advertisements
प्रश्न
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"`
Advertisements
उत्तर
`bar"a".(bar"b" xx bar"c") = |(3,-1, 4),(2,3,-1),(-5,2,3)|`
= 3(9 + 2) +1(6 - 5) + 4(4 + 15)
= 33 + 1 + 76
= 110
APPEARS IN
संबंधित प्रश्न
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]`
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`
Find the value of λ, if four points with position vectors `3hati + 6hatj+9hatk`, `hati + 2hatj + 3hatk`,`2hati + 3hatj + hatk` and `4hati + 6hatj + lambdahatk` are coplanar.
if `bara = 3hati - 2hatj+7hatk`, `barb = 5hati + hatj -2hatk`and `barc = hati + hatj - hatk` then find `bara.(barbxxbarc)`
Show that the four points A, B, C and D with position vectors `4hati + 5hatj + hatk`, `-hatj-hatk`, `3hati + 9hatj + 4hatk` and `4(-hati + hatj + hatk)` respectively are coplanar
Give a condition that three vectors \[\vec{a}\], \[\vec{b}\] and \[\vec{c}\] form the three sides of a triangle. What are the other possibilities?
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 \[l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\]
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:
\[\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} , \vec{b} =\hat{ i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} + 2 \hat{k}\]
Show of the following triad of vector is coplanar:
\[\vec{a} = - 4 \hat{i} - 6 \hat{j} - 2 \hat{k} , \vec{b} = -\hat{ i} + 4 \hat{j} + 3 \hat{k} , \vec{c} = - 8 \hat{i} - \hat{j} + 3 \hat{k}\]
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:
\[\vec{a} = \hat {i} + 3 \hat {j} , \vec{b} = 5 \hat {k} , \vec{c} = \lambda \hat {i} - \hat {j}\]
Show that four points whose position vectors are
\[6 \hat { i} - 7 \hat {j} , 16 \hat { i} - 19 \hat { j} - 4 \hat {k} , 3 \hat {i} - 6 \hat {k} , 2 \hat { i} - 5 \hat {j}+ 10 \hat {k}\]
Find the value of λ for which the four points with position vectors
\[-\hat { j} - \hat {k} , 4 \hat {i} + 5 \hat {j} + \lambda \hat {k} , 3 \hat {i} + 9 \hat {j} + 4 \hat {k} \text { and } - 4 \hat {i} + 4 \hat {j} + 4 \hat{k}\]
\[\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{ and } \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},\]
If c2 = −1 and c3 = 1, show that no value of c1 can make \[\vec{a,} \vec{b}\text { and } \vec{c}\] coplanar.
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 the vectors (sec2 A) \[\hat {i} + \hat {j} + \hat {k} , \hat {i} + \left( \sec^2 B \right) \hat {j} + \hat {k} , \hat {i} + \hat {j} + \left( \sec^2 C \right) \hat {k}\] are coplanar, then find the value of cosec2 A + cosec2 B + cosec2 C.
If \[\vec{a,} \vec{b,} \vec{c}\] are non-coplanar vectors, then find the value of \[\frac{\vec{a} \cdot \left( \vec{b} \times \vec{c} \right)}{\left( \vec{c} \times \vec{a} \right) \cdot \vec{b}} + \frac{\vec{b} \cdot \left( \vec{a} \times \vec{c} \right)}{\vec{c} \cdot \left( \vec{a} \times \vec{b} \right)}\].
The value of \[\left[ \vec{a} - \vec{b} , \vec{b} - \vec{c} , \vec{c} - \vec{a} \right], \text { where } \left| \vec{a} \right| = 1, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 3, \text { is }\]
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 \[\vec{a} = 2\hat{ i} - 3 \hat { j} + 5 \hat { k} , \vec{b} = 3 \hat {i} - 4 \hat {j} + 5 \hat {k} \text { and } \vec{c} = 5\hat { i } - 3 \hat {j}- 2 \hat{k},\] then the volume of the parallelopiped with conterminous edges \[\vec{a} + \vec{b,} \vec{b} + \vec{c,} \vec{c} + \vec{a}\] is
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 =
If \[\vec{a,} \vec{b,} \vec{c}\] are three non-coplanar vectors, then \[\left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \left( \vec{a} + \vec{b} \right) \times \left( \vec{a} + \vec{c} \right) \right]\] equals
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 `bb(bara)` and `bb(barb)` are orthogonal, parallel or neither.
`bara = -3/5hati + 1/2hatj + 1/3hatk , barb = 5hati + 4hatj + 3hatk`
Prove by vector method, that the angle subtended on semicircle is a right angle.
If a vector has direction angles 45° and 60°, find the third direction angle.
Find the angle between the lines whose direction cosines l, m, n satisfy the equations 5l + m + 3n = 0 and 5mn − 2nl + 6lm = 0.
Using properties of scalar triple product, prove that `[(bara + barb, barb + barc, barc + bara)] = 2[(bara, barb, barc)]`.
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors `- 6hat"i" + 14hat"j" + 10hat"k", 14hat"i" - 10hat"j" - 6hat"k"` and `2hat"i" + 4hat"j" - 2hat"k"`
Ler `vec"a" = hat"i" + hat"j" + hat"k", vec"b" = hat"i"` and `vec"c" = "c"_1hat"i" + "c"_2hat"j" + "c"_3hat"k"`. If c1 = 1 and c2 = 2. find c3 such that `vec"a", vec"b"` and `vec"c"` are coplanar
If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k", hat"i" + hat"k"` and `"c"hat"i" + "c"hat"j" + "b"hat"k"` are coplanar, prove that c is the geometric mean of a and b
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
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 ______.
Prove that the volume of a tetrahedron with coterminus edges `overlinea, overlineb` and `overlinec` is `1/6[(overlinea, overlineb, overlinec)]`.
Hence, find the volume of tetrahedron whose coterminus edges are `overlinea = hati + 2hatj + 3hatk, overlineb = -hati + hatj + 2hatk` and `overlinec = 2hati + hatj + 4hatk`.
Determine whether `bara and barb` are 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.
`bar a = -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`.
Determine whether `bb(bara and barb)` are orthogonal, parallel or neither.
`bara=-3/5hati+1/2hatj+1/3hatk,barb=5hati+4hatj+3hatk`
Determine whether `bara and barb` is orthogonal, parallel or neither.
`bara = -3/5hati + 1/2hatj + 1/3hatk, barb = 5hati + 4hatj + 3hatk`
If `baru = hati - 2hatj + hatk, barv = 3hati + hatk "and" barw = hatj - hatk` are given vectors, then find `[baru + barw]·[(baru xx barv)xx(barv xx barw)]`
