Vector Algebra
Cross Product
Grade None
Question:
<p>[JEE Main 2019] Let \(\vec{a}=\hat{i}-\hat{j}\) and \(\vec{b}=-\hat{i}+\hat{j}+\hat{k}\) be two given vectors. Let \(\vec{c}=\vec{a}\times\vec{b}\). Then which of the following is NOT true?</p>
\(|\vec{c}|=\sqrt{3}\) and \(\vec{c}\cdot\vec{a}=0\)
\(\vec{c}\cdot\vec{b}=0\) but \(\vec{c}\cdot\vec{a}\neq0\)
\(\vec{c}\perp\vec{a}\) and \(\vec{c}\perp\vec{b}\)
\(|\vec{c}|=|\vec{a}||\vec{b}|\sin(\text{angle})\)
Step-by-Step Solution
Key Concept: c=a \times b. Cross product is perpendicular to both a and b. Compute magnitude and verify.
$\vec{a}=\hat{i}-\hat{j}+0\hat{k},\;\vec{b}=-\hat{i}+\hat{j}+\hat{k}$.
$\vec{c}=\vec{a}\times\vec{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\1&-1&0\\-1&1&1\end{vmatrix}=(-1-0)\hat{i}-(1-0)\hat{j}+(1-1)\hat{k}=-\hat{i}-\hat{j}$.
$|\vec{c}|=\sqrt{1+1}=\sqrt2\neq\sqrt3$. So option A is NOT true. Answer: (A)
Correct Answer: A