Exercise · ZW-DOT-PRODUCT-009 · ★★★ Advanced
If \(u\cdot v=u\cdot w\) and \(u\neq 0\), what can be concluded?
Choose one option.
Show answer
- A. Only that \(u\cdot(v-w)=0\); not necessarily \(v=w\)
Explanation
Subtracting gives \(u\cdot(v-w)=0\), which does not force \(v-w=0\): with \(u=(1,0)\), \(v=(2,5)\), \(w=(2,-3)\) we get \(u\cdot v=u\cdot w=2\) but \(v\neq w\). There is no cancelling \(u\).
Review the concept: Dot product
The dot product of two vectors in \(\mathbb{R}^n\) is the number obtained by multiplying their coordinates one by one and adding the results.
\[u\cdot v=u_1v_1+u_2v_2+\dots+u_nv_n\]
More practice problems: dot product
- ZW-DOT-PRODUCT-001 Compute (1,2,3)·(4,-5,6).
- ZW-DOT-PRODUCT-002 Compute (2,-1,0,3)·(1,4,5,-2).
- ZW-DOT-PRODUCT-003 The dot product of two vectors in ℝ^n is:
- ZW-DOT-PRODUCT-004 Compute (7,-2,5)·(1,0,0).
- ZW-DOT-PRODUCT-005 Compute (3,-4,12)·(3,-4,12).
- ZW-DOT-PRODUCT-006 For which k does (k,2,-1)·(3,k,4)=0 hold?
- ZW-DOT-PRODUCT-007 If u· v=5 and u· w=-2, what is u·(2v-3w)?
- ZW-DOT-PRODUCT-008 Which of these statements is false?