Answer:
A = 21 î + 28 ĵ
B = 12 î + 5 ĵ
Step-by-step explanation:
a.
To write A and B in terms of î and ĵ, we need to use the trigonometric ratios and the vector notation
According to the diagram, we have:
tan a = 4/3 tan ß = 5/12
Using the identity tan θ = opposite/adjacent, we can find the x and y components of A and B.
For A, we have:
x component = 35 cos a y component = 35 sin a
Using tan a = 4/3, we can find cos a and sin a by using Pythagoras’ theorem:
cos a = 3/5 sin a = 4/5
Therefore, the x and y components of A are:
x component = 35 cos a = 35 (3/5) = 21 y component = 35 sin a = 35 (4/5) = 28
Using the vector notation, we can write A as:
A = 21 î + 28 ĵ
Similarly, for B, we have:
x component = 13 cos ß y component = 13 sin ß
Using tan ß = 5/12, we can find cos ß and sin ß by using Pythagoras’ theorem:
cos ß = 12/13 sin ß = 5/13
Therefore, the x and y components of B are:
x component = 13 cos ß = 13 (12/13) = 12 y component = 13 sin ß = 13 (5/13) = 5
Using the vector notation, we can write B as:
B = 12 î + 5 ĵ
b.
To show that A + B makes an angle of 45° to the x-axis, we need to find the resultant vector R and its angle θ with the x-axis.
To find R, we can use the vector addition rule :
R = A + B R = (21 î + 28 ĵ) + (12 î + 5 ĵ) R = (21 + 12) î + (28 + 5) ĵ R = 33 î + 33 ĵ
To find θ, we can use the inverse tangent function :
tan θ = y component / x component tan θ = 33 / 33 tan θ = 1
θ = tan^-1(1) θ = 45°
Therefore, A + B makes an angle of 45° to the x-axis.