Question

Find the dot product of the following vectors
r1=2i+3j−5k,r2=i−2j+4k.Find r1.r2

-24
20
34
23

Answer 1

Answer on Question #47027 – Math – Vector Calculus

Find the dot product of the following vectors.

$$r_1 = 2i + 3j - 5k, \quad r_2 = i - 2j + 4k. \quad \text{Find } r_1 \cdot r_2.$$

-24

20

34

23

Solution:

Points in this 3-dimensional space must therefore have three coordinates, not two, and are written as ordered triples: $(x, y, z)$. Similarly, vectors will now have three components, such that vector $r_1$ will have components $r_{1x}, r_{1y}$ and $r_{1z}$. Writing in $ijk$ notation, we then have $k$, the unit vector pointing along the $z$-direction. Three-dimensional vectors can also be written with magnitudes and directions. We can now state the definition of the dot product in 3D form:

$$r_1 \cdot r_2 = r_{1x}, r_{2x} + r_{1y}, r_{2y} + r_{1z}, r_{2z}$$

Now we can determine the dot product of the given vectors. We apply the formula noted above.

The dot product of $r_1 \cdot r_2$ equals

$$r_1 \cdot r_2 = 2 \cdot 1 + 3 \cdot (-2) + (-5) \cdot 4 = 2 + (-6) + (-20) = -4 - 20 = -24$$

Answer: The dot product of the following vectors $r_1$ and $r_2$ is equal to -24.

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