(b) Vector Triple Product
Consider next the cross product of
This is a vector perpendicular to both a is normal to the plane of
so
must lie in this plane. It is therefore expressible in terms of
in the form
To find the actual expression for
consider unit vectors j^ and k^ the first parallel to
and the second perpendicular to it in the plane
In terms of j^ and k^ and the other unit vector î of the right-handed system, the remaining vector be written
Then
and the triple product
This is the required expression for in terms of
Similarly the triple product ...(2)
It will be noticed that the expansions (1) and (2) are both written down by the same rule. Each scalar product involves the factor outside the bracket; and the first is the scalar product of the extremes.
In a vector triple product the position of the brackets cannot be changed without altering the value of the product. For is a vector expressible in terms of
is one expressible in terms of
The products in general therefore represent different vectors. If a vector r is resolved into two others in the plane of
one parallel to and the other perpendicular to it, the former is
and therefore the latter
Geometrical Interpretation of
Consider the expression which itself is a vector, since it is a cross product of two vectors
Now
is a vector perpendicular to the plane containing
vector perpendicular to the plane
therefore
is a vector lies in the plane of
and perpendicular to a . Hence we can express
in terms of
i.e.
where x & y are scalars.
Ex.24 Find a vector and is orthogonal to the vector
It is given that the projection of
Sol.
A vector coplanar with is parallel to the triple product,
Ex.25 ABCD is a tetrahedron with A(–5, 22, 5); B(1, 2, 3); C(4, 3, 2); D(–1, 2, –3). Find What can you say about the values of
Calculate the volume of the tetrahedron ABCD and the vector area of the triangle AEF where the quadrilateral ABDE and quadrilateral ABCF are parallelograms.
Sol.
206 videos|263 docs|139 tests
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1. What is a vector triple product? | ![]() |
2. How is the vector triple product calculated? | ![]() |
3. What is the geometric interpretation of the vector triple product? | ![]() |
4. What are some applications of the vector triple product? | ![]() |
5. Can the vector triple product be commutative? | ![]() |
206 videos|263 docs|139 tests
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