Find the dot product of the vectors v 8i 2j and w I 3j F

Find the dot product of the vectors v = 8i - 2j and w = I - 3j. Find the magnitude and direction angle of v = -4(cos30degreei + sin30degreej). Solve the triangleABC with A = 60degree, B = 56degree and side c = l00meters

Solution

Q1)
A) V= 8i -2j and W = i - 3j

Dot product of V.W = (8i - 2j) . (i - 3j)
(8.1) + (-2)(-3) = 8 + 6 = 14.
The dot product of the vectors is 14.

B) Magnitude and direction angle of V = -4 (Cos30i + Sin30j)

Magnitude = ||V|| = Sqrt ( x² + y²) = ((16X3/4) + (16/4)) = Sqrt (16) = 4.
Magnitude = ||V|| = 4.

Direction angle of V = Tan (alpha) = Sin30 / Cos 30 = Tan 30, so Alpha = 30⁰ or Pi/6

Q2) Given
<A = 60⁰ ; <B =56⁰ and Side C = 100m
<C = (180-(60 +56)) = 64⁰.
Angle C =64⁰

We know that SinA/a = SinB/b = SinC/c

SinC/c = SinB/b = Sin 64⁰ /100 = Sin56⁰ / b
Substituting the values , we get Side b = 92.31m

Substitutung the values in SinA/a = Sin C/c, we get the value of Side a = 96.43m