Question

If four distinct points with position vectors \(\overrightarrow a,\overrightarrow b,\overrightarrow c\)  and \(\overrightarrow d\)  are coplanar, then \([\overrightarrow a \overrightarrow b \overrightarrow c]\) is equal to

• A. \([\overrightarrow a \overrightarrow d \overrightarrow b]+[\overrightarrow d \overrightarrow c \overrightarrow a]+[\overrightarrow d\overrightarrow b\overrightarrow c] \)
• B. \([\overrightarrow b \overrightarrow c \overrightarrow d]+[\overrightarrow d \overrightarrow a \overrightarrow c]+[\overrightarrow d\overrightarrow b\overrightarrow a]\)
• C. \([\overrightarrow d \overrightarrow b \overrightarrow a]+[\overrightarrow a \overrightarrow c \overrightarrow c]+[\overrightarrow d\overrightarrow b\overrightarrow c]\)
• D. \([\overrightarrow d \overrightarrow c \overrightarrow a]+[\overrightarrow b \overrightarrow d \overrightarrow a]+[\overrightarrow c\overrightarrow d\overrightarrow b]\)

Answer 1

The Correct Option is D

To determine if four distinct points with position vectors \(\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}, \overrightarrow{d}\) are coplanar, we use the concept of the scalar triple product. The scalar triple product \([\overrightarrow{u} \ \overrightarrow{v} \ \overrightarrow{w}]\) of vectors \(\overrightarrow{u}, \overrightarrow{v}, \overrightarrow{w}\) is defined as:

\([\overrightarrow{u} \ \overrightarrow{v} \ \overrightarrow{w}] = \overrightarrow{u} \cdot (\overrightarrow{v} \times \overrightarrow{w})\)

For four vectors to be coplanar, the volume of the parallelepiped they form must be zero. This condition is given mathematically as:

\([\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c} \ \overrightarrow{d}]\ = [\overrightarrow{d} \ \overrightarrow{c} \ \overrightarrow{a}] + [\overrightarrow{b} \ \overrightarrow{d} \ \overrightarrow{a}] + [\overrightarrow{c} \ \overrightarrow{d} \ \overrightarrow{b}]\ = 0\)

Based on the given options, we need to find how the scalar triple product \([\overrightarrow{a} \ \overrightarrow{b} \ \overrightarrow{c}]\) relates to them. Since the points are coplanar, the correct expression for zero volume is given by cyclic permutations and choices of the vectors:

• The expression leads us to consider permutations of the vectors arranged to keep all the determinant expansions to zero, which leads us to the correct formula.

Hence, the correct option expressing the sum of volume expressions that equal zero due to coplanarity is:

\([\overrightarrow{d} \ \overrightarrow{c} \ \overrightarrow{a}] + [\overrightarrow{b} \ \overrightarrow{d} \ \overrightarrow{a}] + [\overrightarrow{c} \ \overrightarrow{d} \ \overrightarrow{b}]\)

This confirms that the points are coplanar since the scalar triple products sum to zero. Therefore, the option matches our result for the condition of coplanarity.