Vectors in the Plane and Space

≈ 20 min

A vector is an ordered list of numbers carrying both size and direction. In the plane we write $\vec{v}=(v_1,v_2)$ ; in space, $(v_1,v_2,v_3)$ .

Addition is component-wise: $(a,b)+(c,d)=(a+c,\,b+d)$ . Scalar multiplication stretches: $k(a,b)=(ka,kb)$ .

The magnitude (length) comes from Pythagoras:

\|\vec{v}\| = \sqrt{v_1^2 + v_2^2}

The dot product combines two vectors into a single number:

(a,b)\cdot(c,d) = ac + bd

It measures how much two vectors point the same way. A dot product of zero means the vectors are perpendicular.

Worked example. $(2,3)\cdot(4,-1) = 2\cdot 4 + 3\cdot(-1) = 8 - 3 = 5.$

Find the magnitude of the vector $(3,4)$ .

Compute $(1,2)+(3,-1)$ .

$(4,1)$

$(3,-2)$

$(4,3)$

$(2,-3)$

Compute the dot product $(2,3)\cdot(4,-1)$ .

Find the magnitude of the three-dimensional vector $(2,-1,2)$ .