Advertisements
Advertisements
प्रश्न
Determine whether the three vectors `2hat"i" + 3hat"j" + hat"k", hat"i" - 2hat"j" + 2hat"k"` and `3hat"i" + hat"j" + 3hat"k"` are coplanar
Advertisements
उत्तर
Let `vec"a" = 2vec"i" + vec"j" + vec"k"`
`vec"b" = vec"i" - 2vec"j" + 2vec"k"`
`vec"c" = 3vec"i" + vec"j" + 3vec"k"`
We know that `vec"a", vec"b", vec"c"` are coplanar if and only if `[vec"a", vec"b", vec"c"]` = 0
`[vec"a", vec"b", vec"c"] = |(2, 3, 1),(1, -2, 2),(3, 1, 3)|`
= `2(- 6 - 2) - 3(3 - 6) + (1 + 6)`
= −16 + 9 + 7
= 0
∴ Gives vectors are coplanar
APPEARS IN
संबंधित प्रश्न
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
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
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`
Find the volume of the parallelopiped whose coterminus edges are given by vectors `2hati+5hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`
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:
\[\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} , \vec{b} =\hat{ i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} + 2 \hat{k}\]
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 the points A (−1, 4, −3), B (3, 2, −5), C (−3, 8, −5) and D (−3, 2, 1) are coplanar.
Write the value of \[\left[ 2 \hat { i } \ 3 \hat { j }\ 4 \hat { k } \right] .\]
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
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{b} + \vec{c} \right) \times \left( \vec{a} + \vec{b} + \vec{c} \right) =\]
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.
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" + bar"w").[(bar"u" xx bar"v") xx (bar"v" xx bar"w")]`
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 θ is the angle between the unit vectors `bar"a"` and `bar"b"`, the `cos theta = theta/2` = ______.
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` are orthogonal, parallel or neither.
`bara = -3/5hati + 1/2hatj +1/3 hatk, barb = 5hati + 4hatj +3hatk`
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)]`
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).
