Advertisements
Advertisements
Question
Determine where `bb(bara)` and `bb(barb)` are orthogonal, parallel or neither.
`bara = -3/5hati + 1/2hatj + 1/3hatk , barb = 5hati + 4hatj + 3hatk`
Advertisements
Solution
`bara . barb = (-3/5hati + 1/2hatj + 1/3hatk).(5hati + 4hatj + 3hatk)`
= `(-3/5)(5) + (1/2)(4) + (1/3)(3)`
= −3 + 2 + 1
= 0
Since, `bara, barb` are non-zero vectors and `bara.barb` = 0.
`bara` is orthogonal to `barb`.
APPEARS IN
RELATED QUESTIONS
If `bar c = 3bara- 2bar b ` Prove that `[bar a bar b barc]=0`
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`
Let `veca = hati + hatj + hatk = hati` and `vecc = c_1veci + c_2hatj + c_3hatk` then
1) Let `c_1 = 1` and `c_2 = 2`, find `c_3` which makes `veca, vecb "and" vecc`coplanar
2) if `c_2 = -1` and `c_3 = 1`, show that no value of `c_1`can make `veca, vecb and vecc` 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 \[\left[ \vec{a} \vec{b} \vec{c} \right]\] , when \[\vec{a} =\hat{ i} - 2 \hat{j} + 3 \hat{k} , \vec{b} = 2 \hat{i} + \hat{j} - \hat{k}\text{ and } \vec{c} = \hat{j} + \hat{k}\]
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:
\[\hat{a} = \hat{i} - 2 \hat {j} + 3 \hat {k} , \hat {b} = - 2 \hat {i} + 3 \hat {j} - 4 \hat { k}, \hat {c} = \hat { i} - 3 \hat { j} + 5 \hat { k }\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = \hat{i} - \hat{j} + \hat{k} , \vec{b} = 2 \hat {i} + \hat {j} - \hat {k} , \vec{c} = \lambda\hat { i} - \hat {j} + \lambda \hat {k}\]
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
\[-\hat { j} - \hat {k} , 4 \hat {i} + 5 \hat {j} + \lambda \hat {k} , 3 \hat {i} + 9 \hat {j} + 4 \hat {k} \text { and } - 4 \hat {i} + 4 \hat {j} + 4 \hat{k}\]
\[\text {Let } \vec{a} = \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} \text{ and } \hat {c} = c_1 \hat{i} + c_2 \hat {j} + c_3 \hat {k} . \text {Then},\]
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 \[\left[ 2 \hat { i } \ 3 \hat { j }\ 4 \hat { k } \right] .\]
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 \[\left[ \vec{a} - \vec{b} , \vec{b} - \vec{c} , \vec{c} - \vec{a} \right], \text { where } \left| \vec{a} \right| = 1, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 3, \text { is }\]
If \[\vec{a,} \vec{b,} \vec{c}\] are three non-coplanar mutually perpendicular unit vectors, then \[\left[ \vec{a} \vec{b} \vec{c} \right],\] 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 \[\vec{a} = 2\hat{ i} - 3 \hat { j} + 5 \hat { k} , \vec{b} = 3 \hat {i} - 4 \hat {j} + 5 \hat {k} \text { and } \vec{c} = 5\hat { i } - 3 \hat {j}- 2 \hat{k},\] then the volume of the parallelopiped with conterminous edges \[\vec{a} + \vec{b,} \vec{b} + \vec{c,} \vec{c} + \vec{a}\] is
If \[\left[ 2 \vec{a} + 4 \vec{b} \vec{c} \vec{d} \right] = \lambda\left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right],\] then λ + μ =
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 =
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"`
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"`
If a line has the direction ratios 4, −12, 18, then find its direction cosines
If the vectors `- 3hati + 4hatj - 2hatk , hati + 2hatk` and `hati - phatj` are coplanar, then find the value of p.
Using properties of scalar triple product, prove that `[(bara + barb, barb + barc, barc + bara)] = 2[(bara, barb, barc)]`.
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
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 `veca = hati + hatj + hatk, veca.vecb` = 1 and `veca xx vecb = hatj - hatk`, then find `|vecb|`.
If the direction cosines of a line are `(1/c, 1/c, 1/c)` then ______.
If `veca, vecb, vecc` are three non-coplanar vectors, then the value of `(veca.(vecb xx vecc))/((vecc xx veca).vecb) + (vecb.(veca xx vecc))/(vecc.(veca xx vecb))` 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`
If a vector has direction angles 45ºand 60º find the third direction angle.
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 `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)]`
