Q1.5 Multiplying vectors: the dot (or scalar) product (S)
Write down the definition of the dot product of two vectors.
The three vectors
,
and
form a right-angled triangle
in which B=2 and C=1 (in arbitrary units).
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Solution
Reveal
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and
here– is (Key Point 0.3)
where θ is the angle between the two vectors.
Solution
The definition of the dot product of two vectors –let’s call themIt follows that in this case
while![\[ \vec{A}\cdot\vec{B} = AB \cos {\theta_{AB}} = AB \times\frac{B}{A} =B^2 = 4 \]](mastermathpng-10.png)
![\[ \vec{A}\cdot\vec{C} = AC \cos {\theta_{AC}} = AC \times[-\frac{C}{A}] =-C^2 =-1 \]](mastermathpng-11.png)
The minus sign comes from the fact that the angle between
(you need to extend it) and
is greater than 90∘.- From the definition it follows that
is the same as
![\[ \vec{A}\cdot\vec{B} = \mbox{\rm magnitude of A} \times \mbox{\rm magnitude of B} \times \mbox{\rm cos of angle between them} \]](mastermathpng-14.png)
. The order of the vectors in a scalar
product doesn’t matter. - Finally the geometric interpretation of
is that it gives
the projection of one vector (say the first) onto the second, times the
magnitude of the second. In this case the projection of
on
is just B so that
is indeed B2. A similar argument
recovers our result for 
