Vector Algebra
Statements About Vectors
Grade 12
Question:
<p>For any vector \(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\) with \(10|\vec{a}|\geq1\),
consider:</p>
<p>(A) \(|\vec{a}\times\hat{i}|^2+|\vec{a}\times\hat{j}|^2+|\vec{a}\times\hat{k}|^2=2|\vec{a}|^2\).</p>
<p>(B) \(|\vec{a}\cdot\hat{i}|^2+|\vec{a}\cdot\hat{j}|^2+|\vec{a}\cdot\hat{k}|^2=|\vec{a}|^2\).</p>
<p>Which is true?</p>
<li>Only (A)</li>
<li>Only (B)</li>
<li>Both (A) and (B)</li>
<li>Neither</li>
Step-by-Step Solution
Key Concept: |a \times i|^2 + |a \times j|^2 + |a \times k|^2 = 2|a|^2 (each cross gives two components). |a \cdot i|^2 + |a \cdot j|^2 + |a \cdot k|^2 = a_1^2+a_2^2+a_3^2 = |a|^2.
Statement (A): $|\vec{a}\times\hat{i}|^2=|a_3\hat{j}-a_2\hat{k}|^2=a_2^2+a_3^2$.
Sum over all three: $(a_2^2+a_3^2)+(a_1^2+a_3^2)+(a_1^2+a_2^2)=2(a_1^2+a_2^2+a_3^2)=2|\vec{a}|^2$. ✓
Statement (B): $|\vec{a}\cdot\hat{i}|^2+|\vec{a}\cdot\hat{j}|^2+|\vec{a}\cdot\hat{k}|^2
=a_1^2+a_2^2+a_3^2=|\vec{a}|^2$. ✓
Both are true: answer C .
Correct Answer: C