Tricky dot product 2

Geometry Level 1

True or False?

If $\vec{a}$ and $\vec{b}$ are vectors such that $\vec{a} \cdot \vec{b} = 1$ , then $\vec{a}$ and $\vec{b}$ must be parallel.

True False

3 solutions

Brian Charlesworth
Nov 9, 2015

Counterexample: Let $\vec a = <3, -2>$ and $\vec b = <1,1>.$ Then $\vec a \cdot \vec b = 3 \times 1 + (-2) \times 1 = 1,$

but $<3,-2> \ne k<1,1>$ for all real $k,$ and thus the two vectors are not parallel.

Brian, could you explain a bit more simply, please? This is the first page I have ever read about vectors in my life. I've got no idea what that funny star-shaped symbol means.

Sapere Aude - 2 years, 5 months ago

The "funny star-shaped symbol" was just a multiplication symbol; for sake of clarity I've just now changed it to $\times$ . The calculation made here is called the dot product; for an introduction to this you can check out the subsection of this wiki .

Brian Charlesworth - 2 years, 5 months ago

Mahmudul Kabir Rafi
Oct 30, 2019

If a and b must be parallel then the angle between them is either 0 degree or 180 degree. we know that the value of cos0 is 1 and cos180 is -1.but magnitude of a and b are not always 1.hence the answer is false.

Shubhrajit Sadhukhan
May 4, 2021

Counterexample:

Let $a = 1, b = 2, θ = 60°$

$\vec a \cdot \vec b = ab \cosθ = 1$

Tricky dot product 2

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