Vector Algebra
Scalar Triple Product
Grade 12
Question:
<p>Let <i>r</i>, <i>a</i>, <i>b</i> and <i>c</i> be four non-zero vectors such that <i>r</i> · <i>a</i> = 0, |<i>r</i> × <i>b</i>| = |<i>r</i>| |<i>b</i>|, |<i>r</i> × <i>c</i>| = |<i>r</i>| |<i>c</i>|, then [<i>a b c</i>] is</p>
<p>(a) |<i>a</i>| |<i>b</i>| |<i>c</i>|</p>
<p>(b) −|<i>a</i>| |<i>b</i>| |<i>c</i>|</p>
<p>(c) 0</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: If a vector is perpendicular to three coplanar vectors, the scalar triple product must be zero.
Given: r · a = 0 From | r × b | = | r | | b |: r is perpendicular to b From | r × c | = | r | | c |: r is perpendicular to c Conclusion: r is perpendicular to a , b , and c Therefore, r is perpendicular to the plane containing a , b , and c , which means [ a b c ] = 0 ∴ Answer is (c) 0
Correct Answer: C