Question

For $\boldsymbol{a}=(1,3,-2), \boldsymbol{b}=(0,3,1), \boldsymbol{c}=(0,-1,2)$, find:$$ a \times c+c \times a $$

   For $\boldsymbol{a}=(1,3,-2), \boldsymbol{b}=(0,3,1), \boldsymbol{c}=(0,-1,2)$, find:$$
a \times c+c \times a
$$

The Chemistry Maths Book

The Chemistry Maths Book

Erich Steiner 2nd Edition

Chapter 16, Problem 34

Instant Answer

Solved by Expert Adriano Chikande

01/13/2022

Step 1

The cross product of two vectors $\boldsymbol{a}=(a_1,a_2,a_3)$ and $\boldsymbol{c}=(c_1,c_2,c_3)$ is given by: $$ \boldsymbol{a} \times \boldsymbol{c} = (a_2c_3 - a_3c_2, a_3c_1 - a_1c_3, a_1c_2 - a_2c_1) $$ Substituting the given values, we Show more…

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For $\boldsymbol{a}=(1,3,-2), \boldsymbol{b}=(0,3,1), \boldsymbol{c}=(0,-1,2)$, find:$$ a \times c+c \times a $$

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Key Concepts

Cross Product

The cross product is a binary operation on two three-dimensional vectors that results in another vector perpendicular to the plane containing the two original vectors. Its magnitude is given by the area of the parallelogram that the two vectors span, and its direction is determined using the right-hand rule.

Anti-Commutativity of the Cross Product

The cross product is anti-commutative, meaning that swapping the order of the vectors reverses the direction of the resulting vector (i.e., u × v = - (v × u)). This property simplifies calculations in vector algebra and is critical when combining cross products, as it can lead to cancellation of terms.

Zero Vector Result from Opposite Cross Products

When the cross product of two vectors and its reverse are added together (such as u × v + v × u), the result is the zero vector. This is a direct consequence of the anti-commutative property and highlights an important aspect of vector operations in three-dimensional space.

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