Fisika SMA - Vektor (4) - Penjumlahan Vektor Metode Analitik, Rumus Penjumlahan Vektor (I)

Summary

The lesson explains how to find the magnitude and direction of a resultant vector using equations, as an alternative to graphical methods such as the triangle and parallelogram methods.

Key formulas

For two vectors (A) and (B), with included angle (\theta):

• Magnitude of their sum [ R_{\text{sum}}=\sqrt{A^2+B^2+2AB\cos\theta} ]

• Magnitude of their difference [ R_{\text{diff}}=\sqrt{A^2+B^2-2AB\cos\theta} ]

Here, (\theta) is the angle between the two vectors, not the angle either vector makes with an axis.

To find the resultant’s direction, the lesson uses the sine rule on the triangle formed by the vectors. If (\alpha) is the angle opposite side (B), and (\beta=180^\circ-\theta), then:

[ \frac{\sin\alpha}{B}=\frac{\sin\beta}{R} ]

The specific angle represented by (\alpha) depends on the diagram.

Method for solving vector-resultant problems

1. Draw a coordinate diagram if the vectors are specified by directions relative to the axes.
2. Determine the included angle between the vectors. Use the straight angle (180^\circ) or right angle (90^\circ) as needed.
3. Substitute the vector magnitudes and included angle into the appropriate sum or difference formula.
4. To find the direction, form the vector triangle and apply the sine rule. Identify which side is opposite the angle being calculated.
5. Check that the final direction is measured from the intended axis.

Worked examples

• Two forces: (F_1=2\text{ N}) at (30^\circ) from the positive (x)-axis and (F_2=4\text{ N}) along the positive (y)-axis have an included angle of (60^\circ). Their resultant magnitude is: [ R=\sqrt{2^2+4^2+2(2)(4)\cos60^\circ} =\sqrt{28}=2\sqrt7\text{ N} ]

• Two momenta: (P_1=12\text{ kg·m/s}) at (37^\circ) from the positive (x)-axis and (P_2=16\text{ kg·m/s}) at (53^\circ) from the negative (x)-axis are at (90^\circ) to one another. Their resultant is: [ R=\sqrt{12^2+16^2}=20\text{ kg·m/s} ]

• Perpendicular forces: Forces of (9\text{ N}) along (+x) and (12\text{ N}) along (+y) give: [ R=\sqrt{9^2+12^2}=15\text{ N} ]

Caption discrepancy: The subtitles report the direction as (37^\circ) from (+x), but their sine-rule calculation gives (\sin\alpha=12/15), corresponding to about (53^\circ) from (+x). The (37^\circ) angle is measured from (+y).

• Equal-magnitude vectors separated by (120^\circ): If both vectors have magnitude (F), their sum has magnitude (F), while their difference has magnitude (\sqrt3F). The ratio of the sum’s magnitude to the difference’s magnitude is: [ 1:\sqrt3 ]

The instructor also reviews standard sine and cosine values for special angles, including (30^\circ), (45^\circ), (60^\circ), and (90^\circ), to support calculations.

Speakers and sources

• One instructor/presenter, associated with the Le GuruLes channel. No other speakers or external sources are clearly featured.