Solve the problem. Let vector u have initial point P1=(0,2) and terminal point P2=(−2,6). Let vector v have initial point Q1=(3,0) and terminal point Q2=(1,4). U and v have the same direction. Find ||u|| and ||v||. Is u=v?

Solve the problem. Let vector u have initial point P1=(0,2) and terminal point P2=(−2,6). Let vector v have initial point Q1=(3,0) and terminal point Q2=(1,4). U and v have the same direction. Find ||u|| and ||v||. Is u=v?

Question
Vectors
asked 2021-02-02
Solve the problem. Let vector u have initial point P1=(0,2) and terminal point P2=(−2,6). Let vector v have initial point Q1=(3,0) and terminal point Q2=(1,4). U and v have the same direction. Find ||u|| and ||v||. Is u=v?

Answers (1)

2021-02-03
If a vector has an initial point of (x1,y1) and a terminal point of (x2,y2), then the component form of the vector is ⟨x2−x1,y2−y1⟩.
If v has an initial point of P1=(0,2) and P2(−2,6), then v=⟨−2−0,6−2⟩=⟨−2,4⟩.
If u has an initial point of Q1=(3,0) and Q2(1,4), then u=⟨1−3,4−0⟩=⟨−2,4⟩.
The magnitude of a vector with component form of ⟨a,b⟩ is \(\displaystyle√{\left({\left({a}^{{2}}\right)}+{\left({b}^{{2}}\right)}\right)}\)
Since uu and vv have the same component form of ⟨−2,4⟩, they have the same magnitude of:
\(\displaystyle∣∣{u}∣∣=∣∣{v}∣∣=√{\left({\left(−{2}\right)}^{{2}}\right)}+{4}^{{2}}=√{\left({4}+{16}\right)}=√{20}={2}√{5}.\)
Two vectors are equal if they have the same direction and magnitude. It is given that they have the same direction and we know they have the same magnitude so we can then conclude that u=v. Two vectors are also equal if they have the same component form so we could have also concluded that u=v since they both have a component form of ⟨−2,4⟩.
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