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If `bara = hati - 2hatj`, `barb = hati + 2hatj, barc = 2hati + hatj - 2hatk`, then find (i) `bara xx (barb xx barc)` (ii) `(bara xx barb) xx barc`. Are the results same? Justify.
Concept: Vector Triple Product
Using properties of scalar triple product, prove that `[(bara + barb, barb + barc, barc + bara)] = 2[(bara, barb, barc)]`.
Concept: Scalar Triple Product
If `bar("a") = 4hat"i" + 3hat"k"` and `bar("b") = -2hat"i" + hat"j" + 5hat"k"`, then find `2bar("a") + 5bar("b")`
Concept: Vector Joining Two Points in Algebra
Prove that the volume of a tetrahedron with coterminus edges `overlinea, overlineb` and `overlinec` is `1/6[(overlinea, overlineb, overlinec)]`.
Hence, find the volume of tetrahedron whose coterminus edges are `overlinea = hati + 2hatj + 3hatk, overlineb = -hati + hatj + 2hatk` and `overlinec = 2hati + hatj + 4hatk`.
Concept: Scalar Triple Product
Concept: Vector Joining Two Points in Algebra
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).
Concept: Scalar Triple Product
If the lines
`(x-1)/-3=(y-2)/(2k)=(z-3)/2 and (x-1)/(3k)=(y-5)/1=(z-6)/-5`
are at right angle then find the value of k
Concept: Shortest Distance Between Two Lines
Find the shortest distance between the lines
`bar r = (4 hat i - hat j) + lambda(hat i + 2 hat j - 3 hat k)`
and
`bar r = (hat i - hat j + 2 hat k) + mu(hat i + 4 hat j -5 hat k)`
where λ and μ are parameters
Concept: Shortest Distance Between Two Lines
Show that the points (1, 1, 1) and (-3, 0, 1) are equidistant from the plane `bar r (3bari+4barj-12bark)+13=0`
Concept: Distance of a Point from a Plane
Show that the lines ` (x+1)/-3=(y-3)/2=(z+2)/1; ` are coplanar. Find the equation of the plane containing them.
Concept: Coplanarity of Two Lines
Find the equation of the planes parallel to the plane x + 2y+ 2z + 8 =0 which are at the distance of 2 units from the point (1,1, 2)
Concept: Distance of a Point from a Plane
Find the shortest distance between the lines `(x+1)/7=(y+1)/(-6)=(z+1)/1 and (x-3)/1=(y-5)/(-2)=(z-7)/1`
Concept: Shortest Distance Between Two Lines
Show that the points (1, –1, 3) and (3, 4, 3) are equidistant from the plane 5x + 2y – 7z + 8 = 0
Concept: Distance of a Point from a Plane
Find the co-ordinates of the point, which divides the line segment joining the points A(2, − 6, 8) and B(− 1, 3, − 4) externally in the ratio 1 : 3.
Concept: Distance in Lines (Point & Parallel Lines)
Find the distance of the point (1, 2, –1) from the plane x - 2y + 4z - 10 = 0 .
Concept: Distance of a Point from a Plane
A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.
Concept: Vector and Cartesian Equations of a Line
Show that the line `(x - 2)/(1) = (y - 4)/(2) = (z + 4)/(-2)` passes through the origin.
Concept: Vector and Cartesian Equations of a Line
Find the co-ordinates of the foot of the perpendicular drawn from the point `2hati - hatj + 5hatk` to the line `barr = (11hati - 2hatj - 8hatk) + λ(10hati - 4hatj - 11hatk).` Also find the length of the perpendicular.
Concept: Equation of a Plane
Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 6y – 3z = 63.
Concept: Equation of a Plane
Find the vector equation of the plane passing through the point having position vector `hati + hatj + hatk` and perpendicular to the vector `4hati + 5hatj + 6hatk`.
Concept: Equation of a Plane
