Show that a. (b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors, a, b and c.

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Solution

A parallelepiped with origin O and sides a, b, and c is shown in the following figure.

Volume of the given parallelepiped = abc

Let be a unit vector perpendicular to both b and c. Hence, and a have the same direction.

∴

= abc cosθ

= abc cos 0°

= abc

= Volume of the parallelepiped

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Similar questions

Given a + b + c + d = 0, which of the following statements are correct:

(a) a, b, c, and d must each be a null vector,

(b) The magnitude of (a + c) equals the magnitude of (b + d),

(d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear?

a = i - j + k, a \cdot b = 0, a \times b = c,

c = - 2 i - j + k

b

Establish the following vector inequalities geometrically or otherwise:

(a) |a + b| ≤ |a| + |b|

(b) |a + b| ≥ ||a| − |b||

(d) |a − b| ≥ ||a| − |b||

When does the equality sign above apply?

\begin{matrix} a & a & a \\ \times & a & a & a \\ a & a & a \\ a & a & a & 0 \\ a & a & a & 0 & 0 \end{matrix} a b c b a

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