Advertisements
Advertisements
प्रश्न
Find the altitude of a parallelepiped determined by the vectors `vec"a" = - 2hat"i" + 5hat"j" + 3hat"k", vec"b" = hat"i" + 3hat"j" - 2hat"k"` and `vec"c" = - vec"i" + vec"j" + 4vec"k"` if the base is taken as the parallelogram determined by `vec"b"` and `vec"c"`
Advertisements
उत्तर
Volume = Base Area × Height
`|[(vec"a", vec"b", vec"c")]| = |vec"b" xx vec"c"| xx "Height"`
`|[(vec"a", vec"b", vec"c")]| = |(-2, 5, 3),(1, 3, -2),(-3, 1, 4)|`
= 2(12 + 2) − 5(4 − 6) + 3(1 + 9)
= − 28 + 10 + 30
= 12
`vec"b" xx vec"c" + |(vec"i", vec"j", vec"k"),(1, 3, -2),(-3, 1, 4)|`
= `vec"i"(14) - vec"j"(- 2) + vec"k"(10)`
`vec"b" xx vec"c" = 14vec"i" + 2vec"j" + 10vec"k"`
`|vec"b" xx vec"c"| = sqrt(196 + 4 + 100_`
= `sqrt(300)`
`(sqrt(300)) xx ("Height")` = 12
Height = `12/sqrt(300)`
= `12/(10sqrt(3))`
= `6/(5sqrt(3)) xx sqrt(3)/sqrt(3)`
= `(6sqrt(3))/(5 xx 3)`
= `(2sqrt(3))/5`
APPEARS IN
संबंधित प्रश्न
Prove that, for any three vector `veca,vecb,vecc [vec a+vec b,vec b+vec c,vecc+veca]=2[veca vecb vecc]`
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).
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
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:
\[\left[ 2 \hat{i}\hat{ j}\ \hat{k}\right] + \left[\hat{i}\hat{ k}\hat {j} \right] + \left[\hat{ k}\hat{ j} 2\hat{ i} \right]\]
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 \[\left[ \vec{a} \vec{b} \vec{c} \right]\] , 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 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:
\[\vec{a} = \hat{i} - \hat{j} + \hat{k} , \vec{b} = 2 \hat {i} + \hat {j} - \hat {k} , \vec{c} = \lambda\hat { i} - \hat {j} + \lambda \hat {k}\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = 2 \hat{i} - \hat {j} + \hat {k} , \vec{b} = \hat {i} + 2 \hat {j} - 3 \hat {k} , \vec{c} = \lambda \hat {i} + \lambda \hat {j} + 5 \hat {k}\]
Show that four points whose position vectors are
\[6 \hat { i} - 7 \hat {j} , 16 \hat { i} - 19 \hat { j} - 4 \hat {k} , 3 \hat {i} - 6 \hat {k} , 2 \hat { i} - 5 \hat {j}+ 10 \hat {k}\]
Write the value of \[\left[ \hat {i} + \hat {j} \ \hat {j} + \hat {k} \ \hat {k} + \hat {i} \right] .\]
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
Show that the vectors `hat (i) - 2 hat(j) + 3 hat (k), - 2 hat(i) + 3 hat(j) - 4 hat(k) " and " hat(i) - 3 hat(j) + 5 hat(k) ` are coplanar.
If the vectors `- 3hati + 4hatj - 2hatk , hati + 2hatk` and `hati - phatj` are coplanar, then find the value of p.
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 `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`
Determine whether `bb(bara and barb)` are orthogonal, parallel or neither.
`bara=-3/5hati+1/2hatj+1/3hatk,barb=5hati+4hatj+3hatk`
