What is the dot product of $< 3, 4, 1 >$ $<3,4,1 >$ and $< 5, - 1, 2 >$ $<5,-1,2 >$ ?

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Answer 1 · 2015-12-19T11:26:39

13

Dot product is distributive.
(e.g. $a \cdot (b + c) = a \cdot b + a \cdot c$ $a*(b+c)=a*b+a*c$ )
Dot product of 2 vectors that are perpendicular is zero
(e.g. $ˆ i \cdot ˆ j = 0$ $hati*hatj=0$ )

$(3 ˆ i + 4 ˆ j + ˆ k) \cdot (5 ˆ i - ˆ j + 2 ˆ k)$ $(3hati+4hatj+hatk) * (5hati-hatj+2hatk)$

$= (3 ˆ i) \cdot (5 ˆ i) + (3 ˆ i) \cdot (- ˆ j) + (3 ˆ i) \cdot (2 ˆ k)$ $= (3hati)*(5hati) + (3hati)*(-hatj) + (3hati)*(2hatk)$

$s + (4 ˆ j) \cdot (5 ˆ i) + (4 ˆ j) \cdot (- ˆ j) + (4 ˆ j) \cdot (2 ˆ k)$ $color(white)(s) + (4hatj)*(5hati) + (4hatj)*(-hatj) + (4hatj)*(2hatk)$

$s + (ˆ k) \cdot (5 ˆ i) + (ˆ k) \cdot (- ˆ j) + (ˆ k) \cdot (2 ˆ k)$ $color(white)(s) + (hatk)*(5hati) + (hatk)*(-hatj) + (hatk)*(2hatk)$

$= 15 + 0 + 0$ $= 15 + 0 + 0$

$s + 0 - 4 + 0$ $color(white)(s) + 0 - 4 + 0$

$s + 0 + 0 + 2$ $color(white)(s) + 0 + 0 + 2$

$= 13$ $= 13$

What is the dot product of <3,4,1><3,4,1 > and <5,−1,2><5,-1,2 >?