If a, b̅, c̅ are three vectors (a̅-b̅) ×(a̅-b̅-c̅) ·(a̅+2 b̅-c̅) is equal to (a) -[ a b̅

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If a, b̅, c̅ are three vectors (a̅-b̅) ×(a̅-b̅-c̅) ·(a̅+2...

If $a, \bar{b}, \bar{c}$ are three vectors $(\bar{a}-\bar{b}) \times(\bar{a}-\bar{b}-\bar{c}) \cdot(\bar{a}+2 \bar{b}-\bar{c})$ is equal to (a) $-\left[\begin{array}{lll}a & \bar{b} & \bar{c}\end{array}\right]$ (b) $\left[\begin{array}{lll}- & \bar{b} & \bar{c}\end{array}\right]$ (c) $2\left[\begin{array}{lll}a & \bar{b} & \bar{c}\end{array}\right]$ (d) $3\left[\begin{array}{l}\bar{a} \overline{\mathrm{b}} \overline{\mathrm{c}}]\end{array}\right]$

If $a, \bar{b}, \bar{c}$ are three vectors $(\bar{a}-\bar{b}) \times(\bar{a}-\bar{b}-\bar{c}) \cdot(\bar{a}+2 \bar{b}-\bar{c})$ is equal to
(a) $-\left[\begin{array}{lll}a & \bar{b} & \bar{c}\end{array}\right]$
(b) $\left[\begin{array}{lll}- & \bar{b} & \bar{c}\end{array}\right]$
(c) $2\left[\begin{array}{lll}a & \bar{b} & \bar{c}\end{array}\right]$
(d) $3\left[\begin{array}{l}\bar{a} \overline{\mathrm{b}} \overline{\mathrm{c}}]\end{array}\right]$

Key Concepts

Distributive Properties in Vector Algebra

The distributive properties of vector addition over the dot and cross products allow complex vector expressions to be broken down into simpler parts. These properties enable the expansion and simplification of expressions involving sums and differences of vectors, which is crucial for manipulation and evaluation in vector algebra problems.

Dot Product

The dot product is a scalar operation that combines two vectors by multiplying their corresponding components and summing the results. It is also expressed as the product of the magnitudes of the two vectors and the cosine of the angle between them. This operation is commutative and distributive over vector addition, and it provides a measure of how much one vector extends in the direction of another.

Scalar Triple Product

The scalar triple product is an operation that takes three vectors and returns a scalar. It is defined as the dot product of one vector with the cross product of the other two. This product represents the volume of the parallelepiped formed by the three vectors and has the property of being invariant under cyclic permutation of the vectors. It is useful in determining the coplanarity of vectors as well as in various geometric and algebraic applications.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space that results in another vector perpendicular to the plane containing the given vectors. Its magnitude equals the area of the parallelogram spanned by the vectors. The direction of the cross product is determined by the right-hand rule, and the operation is anticommutative, meaning that switching the order of the operands changes the sign of the result.

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