Advertisements
Advertisements
Question
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"`
Advertisements
Solution
`bar"a" . bar"b" = (2hat"i" + 3hat"j" - hat"k").(5hat"i" - 2hat"j" + 4hat"k")`
`= (2)(5) + (3)(-2) + (-1)(4)`
= 10 - 6 - 4 = 0
Since, `bar"a",bar"b"` are non-zero vectors and `bar"a".bar"b" = 0`
`bar"a"` is orthogonal to `bar"b"`
APPEARS IN
RELATED QUESTIONS
If `bar c = 3bara- 2bar b ` Prove that `[bar a bar b barc]=0`
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
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 \[l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\]
Show of the following triad of vector is coplanar:
\[\vec{a} = - 4 \hat{i} - 6 \hat{j} - 2 \hat{k} , \vec{b} = -\hat{ i} + 4 \hat{j} + 3 \hat{k} , \vec{c} = - 8 \hat{i} - \hat{j} + 3 \hat{k}\]
Show of the following triad of vector is coplanar:
\[\hat{a} = \hat{i} - 2 \hat {j} + 3 \hat {k} , \hat {b} = - 2 \hat {i} + 3 \hat {j} - 4 \hat { k}, \hat {c} = \hat { i} - 3 \hat { j} + 5 \hat { k }\]
Show that the points A (−1, 4, −3), B (3, 2, −5), C (−3, 8, −5) and D (−3, 2, 1) are coplanar.
Find the value of λ for which the four points with position vectors
\[-\hat { j} - \hat {k} , 4 \hat {i} + 5 \hat {j} + \lambda \hat {k} , 3 \hat {i} + 9 \hat {j} + 4 \hat {k} \text { and } - 4 \hat {i} + 4 \hat {j} + 4 \hat{k}\]
If four points A, B, C and D with position vectors 4 \[\hat { i} +3\] \[\hat { j} +3\] \[\hat { k} ,5\] \[\hat { i} +\] \[x\hat { j} +7\] \[\hat { k} ,5\] \[\hat { i} +3\] \[\hat { j}\] and \[7 \hat{i} + 6 \hat{j} + \hat{k}\] respectively are coplanar, then find the value of x.
Write the value of \[\left[ 2 \hat { i } \ 3 \hat { j }\ 4 \hat { k } \right] .\]
If \[\left[ 3 \vec{a} + 7 \vec{b} \vec{c} \vec{d} \right] = \lambda\left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right],\] then find the value of λ + μ.
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 \[\vec{a,} \vec{b,} \vec{c}\] are non-coplanar vectors, then \[\frac{\vec{a} \cdot \left( \vec{b} \times \vec{c} \right)}{\left( \vec{c} \times \vec{a} \right) \cdot \vec{b}} + \frac{\vec{b} \cdot \left( \vec{a} \times \vec{c} \right)}{\vec{c} \cdot \left( \vec{a} \times \vec{b} \right)}\] is equal to
Find the value of p, if the vectors `hat"i" - 2hat"j" + hat"k", 2hat"i" -5hat"j"+"p" hat "k" , 5hat"i" -9hat"j" + 4 hat"k"` are coplanar.
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 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"`
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 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
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 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 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 ______.
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 ______.
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 ______.
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`
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 vertices are A (3, 2, −1), B (−2, 2, −3) C (3, 5, −2) and D (−2, 5, 4).
If a vector has direction angles 45ºand 60º find the third direction angle.
If `u=hati -2hatj + hatk, barr=3hati + hatk and w=hatj, hatk` are given vectors, then find `[baru + barw]. [(barw xx barr)xx(barr xx barw)]`
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 `
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)]`
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).
The magnitude of a vector which is orthogonal to the vector \[\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\] and is coplanar with the vectors \[\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}\] and \[\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}\] is
