We read your photo as:

Calculate the resultant of two vectors V1 = 3 units and V2 = 4 units inclined at angles 30^ , 90^ , and 120^ respectively. Use the formula R = extsqrt(V12 + V22 + 2V1V2 extcos heta).

Physics

ScanSolve AI Answer

Answers

4 parts
a

6.77 units\text{6.77 units}

b

5 units\text{5 units}

c

3.61 units\text{3.61 units}

10

96.42 N\text{96.42 N}

114.90 N\text{114.90 N}

Step-by-step solution

5 steps

Part 9

To find the resultant of two vectors by calculation, use the formula:

R=V12+V22+2V1V2cos⁡θR = \sqrt{V_1^2 + V_2^2 + 2V_1V_2 \cos\theta}

where V1=3V_1 = 3 units, V2=4V_2 = 4 units, and θ\theta is the angle between them.
Note: The "by drawing" part cannot be performed by an AI.

Part a

6.77 units\text{6.77 units}

Inclined at 30∘30^\circ

  1. Substitute the values into the formula.

    R=(3 units)2+(4 units)2+2(3 units)(4 units)cos⁡(30∘)=9 units2+16 units2+24 units2(32)=25 units2+123 units2=25+12×1.73205 units=25+20.7846 units=45.7846 units≈6.77 units\begin{align*} R &= \sqrt{(3 \text{ units})^2 + (4 \text{ units})^2 + 2(3 \text{ units})(4 \text{ units}) \cos(30^\circ)} \\ &= \sqrt{9 \text{ units}^2 + 16 \text{ units}^2 + 24 \text{ units}^2 \left(\frac{\sqrt{3}}{2}\right)} \\ &= \sqrt{25 \text{ units}^2 + 12\sqrt{3} \text{ units}^2} \\ &= \sqrt{25 + 12 \times 1.73205} \text{ units} \\ &= \sqrt{25 + 20.7846} \text{ units} \\ &= \sqrt{45.7846} \text{ units} \\ &\approx 6.77 \text{ units}\end{align*}

    The resultant is 6.77 units\boxed{\text{6.77 units}}.

Part b

5 units\text{5 units}

Inclined at 90∘90^\circ

  1. Substitute the values into the formula.

    R=(3 units)2+(4 units)2+2(3 units)(4 units)cos⁡(90∘)=9 units2+16 units2+24 units2(0)=25 units2=5 units\begin{align*} R &= \sqrt{(3 \text{ units})^2 + (4 \text{ units})^2 + 2(3 \text{ units})(4 \text{ units}) \cos(90^\circ)} \\ &= \sqrt{9 \text{ units}^2 + 16 \text{ units}^2 + 24 \text{ units}^2 (0)} \\ &= \sqrt{25 \text{ units}^2} \\ &= 5 \text{ units}\end{align*}

    The resultant is 5 units\boxed{\text{5 units}}.

Part c

3.61 units\text{3.61 units}

Inclined at 120∘120^\circ

  1. Substitute the values into the formula.

    R=(3 units)2+(4 units)2+2(3 units)(4 units)cos⁡(120∘)=9 units2+16 units2+24 units2(−12)=25 units2−12 units2=13 units2≈3.61 units\begin{align*} R &= \sqrt{(3 \text{ units})^2 + (4 \text{ units})^2 + 2(3 \text{ units})(4 \text{ units}) \cos(120^\circ)} \\ &= \sqrt{9 \text{ units}^2 + 16 \text{ units}^2 + 24 \text{ units}^2 \left(-\frac{1}{2}\right)} \\ &= \sqrt{25 \text{ units}^2 - 12 \text{ units}^2} \\ &= \sqrt{13 \text{ units}^2} \\ &\approx 3.61 \text{ units}\end{align*}

    The resultant is 3.61 units\boxed{\text{3.61 units}}.

Part 10

96.42 N\text{96.42 N}

A component of a vector is its effective value along a specific direction, obtained by projecting the vector onto that direction. For a vector A⃗\vec{A} at an angle θ\theta to the horizontal, its horizontal component is Ax=Acos⁡θA_x = A \cos\theta and its vertical component is Ay=Asin⁡θA_y = A \sin\theta.

Illustration with a diagram:
Imagine a vector originating from the origin of a Cartesian coordinate system. Its horizontal component is the projection onto the x-axis, and its vertical component is the projection onto the y-axis, forming a right-angled triangle with the vector as the hypotenuse.

Calculation of components:
Given force F=150F = 150 N, inclined at θ=50∘\theta = 50^\circ to the horizontal.

  1. Calculate the horizontal component (FxF_x).

    Fx=Fcos⁡θ=150 N×cos⁡(50∘)F_x = F \cos\theta = 150 \text{ N} \times \cos(50^\circ) Fx≈150 N×0.6428F_x \approx 150 \text{ N} \times 0.6428 Fx≈96.42 NF_x \approx 96.42 \text{ N}

    The horizontal component is 96.42 N\boxed{\text{96.42 N}}.

  2. Calculate the vertical component (FyF_y).

    Fy=Fsin⁡θ=150 N×sin⁡(50∘)F_y = F \sin\theta = 150 \text{ N} \times \sin(50^\circ) Fy≈150 N×0.7660F_y \approx 150 \text{ N} \times 0.7660 Fy≈114.90 NF_y \approx 114.90 \text{ N}

    The vertical component is 114.90 N\boxed{\text{114.90 N}}.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Physics Questions

QGiven vectors U(2,1,3) and V(4,−5,1), calculate the dot product, squared magnitude of U, magnitude…QDetermine the units of the expression v2/g and find the resultant vector based on the provided…QCalculate the magnitude of vector (3,4), find the resultant of (2,3) and (4,1), add matrices…QGiven vectors a and b, find the unit vector in the direction of a and the angle between a and b.QFind the unit vector in the direction of the sum of the vectors v1 = i + 4j + 2k, v2 = 3i − 3j − 2k…
See all Physics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp