Advertisements
Advertisements
Question
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`
Advertisements
Solution
Given `vec"c"` is perpendicular to both `vec"a"` and `vec"b"`
So `vec"c"` is parallel to `vec"a" xx vec"b"`
`[(vec"a", vec"b", vec"c")] = vec"a"*(vec"b" xx vec"c")`
`|[(vec"a", vec"b", vec"c")]| = |vec"a"||vec"b" xx vec"c"|`
= `|vec"a"|vec"b"|vec"c"| sin(pi/6)`
`|[(vec"a", vec"b", vec"c")]| = |vec"a"||vec"b"||vec"c"|(1/2)`
Squaring on both sides `[(vec"a", vec"b", vec"c")]^2 = |vec"a"||vec"b"||vec"c"|^2 1/4` ..........`("since" |vec"c"| = 1)`
`[(vec"a", vec"b", vec"c")]^2 = 1/4|vec"a"|^2|vec"b"|^2`
APPEARS IN
RELATED QUESTIONS
Show that the four points A(4, 5, 1), B(0, –1, –1), C(3, 9, 4) and D(–4, 4, 4) are coplanar.
If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`
(A) 100
(B) 101
(C) 110
(D) 109
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`
Show of the following triad of vector is coplanar:
\[\vec{a} = \hat {i} + 2 \hat{j} - \hat {k} , \vec{b} = 3 \hat {i} + 2 \hat{j} + 7 \hat {k} , \vec{c} = 5 \hat {i} + 6 \hat { j} + 5 \hat {k}\]
Show that the points A (−1, 4, −3), B (3, 2, −5), C (−3, 8, −5) and D (−3, 2, 1) are coplanar.
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.
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 \[\vec{a,} \vec{b,} \vec{c}\] 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"`.
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.
If `vec"a" = hat"i" - 2hat"j" + 3hat"k", vec"b" = 2hat"i" + hat"j" - 2hat"k", vec"c" = 3hat"i" + 2hat"j" + hat"k"`, find `vec"a" * (vec"b" xx vec"c")`
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 `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 `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 ______.
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 = -3/5 hati + 1/2 hatj + 1/3 hatk, barb = 5 hati + 4 hatj + 3 hatk`
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
