Q.The value of $\hat{i}\cdot(\hat{j}\times\hat{k})+\hat{j}\cdot(\hat{i}\times\hat{k})+\hat{k}\cdot(\hat{i}\times\hat{j})$ is (A) 0 (B) -1 (C) 1 (D) 3

Concept: Scalar triple product of unit vectors — each term is the volume of a unit cube, i.e., $\pm 1$.

Step 1: Recall that for any orthonormal right-handed triad,

$\hat{j} \times \hat{k} = \hat{i}$, so $\hat{i} \cdot (\hat{j} \times \hat{k}) = \hat{i} \cdot \hat{i} = 1$.

Step 2: Similarly, $\hat{i} \times \hat{k} = -\hat{j}$, so

$\hat{j} \cdot (\hat{i} \times \hat{k}) = \hat{j} \cdot (-\hat{j}) = -1$. …

The problem asks for the value of

This is a sum of three scalar triple products of the standard unit vectors $\hat{i}, \hat{j}, \hat{k}$. The scalar triple product $\vec{a}\cdot(\vec{b}\times\vec{c})$ gives the signed volume of the parallelepiped formed by the three vectors. For orthonormal basis vectors, the cross products are simple: each cross product of two distinct unit vectors gives the third unit vector, up to a sign determined by the right-hand rule.

Let’s evaluate each term step by step.

1. First term: $\hat{i}\cdot(\hat{j}\times\hat{k})$ By the right-hand rule, $\hat{j}\times\hat{k} = \hat{i}$. So

2. Second term: $\hat{j}\cdot(\hat{i}\times\hat{k})$ Here, $\hat{i}\times\hat{k} = -\hat{j}$ (since swapping the order flips the sign: $\hat{k}\times\hat{i} = \hat{j}$, so $\hat{i}\times\hat{k} = -\hat{j}$). Thus

3. Third term: $\hat{k}\cdot(\hat{i}\times\hat{j})$ We have $\hat{i}\times\hat{j} = \hat{k}$, so

Now add them:

$1 + (-1) + 1 = 1.$ …

Method: Evaluating a scalar triple product of the standard unit vectors

Use this whenever an expression is a sum/combination of terms of the form $\vec{u}\cdot(\vec{v}\times\vec{w})$ built from $\hat{i},\hat{j},\hat{k}$ — a scalar triple product (box product).

Steps

Step 1: Recognise each term as a scalar triple product.

$\vec{u}\cdot(\vec{v}\times\vec{w})$ is the signed volume of the box on the three vectors. For the standard basis it can only equal $+1$, $-1$, or $0$.

Step 2: Read off the value from the cyclic order.

The triad follows the cyclic chain $\hat{i}\to\hat{j}\to\hat{k}\to\hat{i}$:

Any anticyclic order (e.g. $\hat{i}\times\hat{k}=-\hat{j}$) flips the sign to $-1$, and any repeated vector gives $0$. …

Common Mistakes

Mistake 1: Assuming all three terms equal $+1$ and answering $3$.

Why it's wrong: only cyclic-order triple products give $+1$; the middle term $\hat{j}\cdot(\hat{i}\times\hat{k})$ uses the anticyclic order $\hat{i}\times\hat{k}=-\hat{j}$, so it equals $-1$. Correct approach: rewrite each cross product in cyclic form and track the sign before adding, giving $1-1+1=1$.

Mistake 2: Writing $\hat{i}\times\hat{k}=\hat{j}$.

Why it's wrong: the cyclic chain is $\hat{i}\to\hat{j}\to\hat{k}\to\hat{i}$; $\hat{i}\times\hat{k}$ runs against it, so $\hat{i}\times\hat{k}=-\hat{j}$. Correct approach: swapping the order of a cross product flips its sign — check the cyclic order every time. …