Vector Algebra
Cross Product and Scalar Triple Product
Grade 12
Question:
<p><strong>Ex. 96</strong> <strong>Statement I:</strong> If <strong>a</strong> is perpendicular to <strong>b</strong> and <strong>c</strong>, then <strong>a</strong> × (<strong>b</strong> × <strong>c</strong>) = 0</p><p><strong>Statement II:</strong> If <strong>b</strong> is perpendicular to <strong>c</strong>, then <strong>b</strong> × <strong>c</strong> = 0</p>
<p>(a) Both Statements are true and Statement II is the correct explanation of Statement I</p>
<p>(b) Both Statements are true but Statement II is not the correct explanation of Statement I</p>
<p>(c) Statement I is true but Statement II is false</p>
<p>(d) Both Statements are false</p>
Step-by-Step Solution
Key Concept: A vector perpendicular to two vectors is parallel to their cross product. The cross product of perpendicular vectors is non-zero.
Step 1: If a is perpendicular to b and c , then a is parallel to ( b × c ). Step 2: Therefore, a × ( b × c ) = 0 (cross product of parallel vectors is zero). Step 3: Statement I is true. Step 4: If b ⊥ c , then b × c ≠ 0 (the cross product of perpendicular non-zero vectors is non-zero). Step 5: Statement II is false. ∴ Answer is (c).
Correct Answer: C