The vectors a and b represent two forces acting at the same point, and θis the smallest positive

step 1 As shown in the figure, first we decompose A and B into their horizontal and vertical components

The vectors a and b represent two forces acting at the same...

The vectors a and b represent two forces acting at the same point, and $\theta$ is the smallest positive angle between a and b. Approximate the magnitude of the resultant force. \begin{aligned} &\text { (a) } 70 \mathrm{Ib}, 320^{\circ}\\ &\text { (b) } 40 \mathrm{lb}, 30^{\circ} \end{aligned} (a) $70 \mathrm{lb}, 320^{\circ}$ (b) $40 \mathrm{lb}, 30^{\circ}$

The vectors a and b represent two forces acting at the same point, and $\theta$ is the smallest positive angle between a and b. Approximate the magnitude of the resultant force.
\begin{aligned}
&\text { (a) } 70 \mathrm{Ib}, 320^{\circ}\\
&\text { (b) } 40 \mathrm{lb}, 30^{\circ}
\end{aligned}
(a) $70 \mathrm{lb}, 320^{\circ}$
(b) $40 \mathrm{lb}, 30^{\circ}$

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Key Concepts

Angle Between Vectors

The angle between two vectors is the smallest positive angle that separates them. This angle is crucial when applying the law of cosines or performing component analysis, as it determines how the vectors interact and combine in terms of magnitude and direction.

Law of Cosines

The law of cosines is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. In vector problems, it is used to calculate the magnitude of the resultant vector when two vectors and the angle between them are known, especially when the vectors are not perpendicular.

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. This is often performed by summing corresponding components or by using graphical methods, and it is fundamental to analyzing forces acting on a body, as it allows for the determination of both the magnitude and the direction of the resulting effect.

Resultant Force

The resultant force is the single force that has the same effect on an object as all the individual forces acting together. It is obtained by vector addition of all the individual forces, and its determination is essential in understanding the overall impact on the motion or equilibrium of the object.

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