The product of scalar multiplied by a scalar

scalar (only magnitude)

Examples: power= E/t or speed= distance/t

Hint: When we ask how fast was the car moving, then we are refering to a scalar quantity which is speed. But, when we add direction to that speed, then we are talking about a vector quantity which has both the direction and the magnitude.

The product of vector multiplied by a scalar

vector dot product/scalar product

Examples: torque= rFSina, or anything with this form ABSin/Cosa

The product of vector multiplied by a vector

vector or scalar vector product or cross product and will always produce a vector or scalar. Magnetic force is the example of vector cross product.

How to identify a scalar quantity

In identifying a scalar quantity, look for a quantity that only has magnitude, but a more surefire method is to find a quantity that consists of a scalar multiplied by a scalar.

capacitance, power, and density.

torque is a vector quantity.

Power is energy divided by time, which is a scalar divided by a scalar. (P=E/t)

How to identify a vector quantity

In identifying a vector quantity, look for a quantity that has both direction and magnitude, but a more surefire method is to find a quantity that consists of a vector multiplied by a scalar.

Force is mass multiplied by acceleration, which is a scalar multiplied by a vector. Hence, it is a vector (F=ma).

Momentum is mass multiplied by velocity, which is a scalar multiplied by a vector. Hence, it is a vector (p=mv).

Electric field Vector or scalar

Electric field is force divided by charge, which is a vector divided by a scalar. Hence, it is a vector (E=F/q).

Other examples: F=ma or P=mv

Electric potential

vector or scalar

Electric potential is electric potential energy divided by charge, which is a scalar divided by a scalar. Hence, it is a **scalar. **

**V= U/q**

The portion of the Mississippi River has a velocity of 1.0 meters per second. In still water, the ferry travels with a velocity of 1.4 meters per second. The ferry must transport the residents directly across the 1600-meter river span.

While this may not be immediately apparent, understand that 1.4 is equal to √2. This creates a relationship for a 45-45-90 triangle, where the ratio of the sides is 1:1:√2. Therefore, the ferry will launch at an angle of 45° from the shore. The ferry must launch upstream at an angle of 45° from the shore with a velocity of 1.4 meters per second.

Given that θ = arctan(1.33), what can be deduced?

One big clue to solving this problem is that the angle is equal to arctan(1.33). 1.33 is the value of the opposite side divided by the adjacent side. Let’s represent the decimal by a fraction, which would be 4/3. So the ratio of the opposite side to the adjacent side is 4:3.

Aisha, Saul, and Lorenzo are playing a tug-of-war. Saul and Lorenzo are pulling at 90 ° to each other, while Aisha is pulling with 20N on her rope at an angle (90 + θ)° from Lorenzo. Given that θ = arctan(1.33), what are the magnitudes of force with which Saul and Lorenzo must pull in order to keep this system in equilibrium? (Figure may not be drawn to scale.)

Aisha is pulling with 20N, so in order for this system to be in equilibrium, Saul and Lorenzo together must be pulling with the same force in the opposite direction. That vector would represent the hypotenuse of the right triangle where Saul’s force is the adjacent side and Lorenzo’s opposite.

One big clue to solving this problem is that the angle is equal to arctan(1.33). 1.33 is the value of the opposite side divided by the adjacent side. Let’s represent the decimal by a fraction, which would be 4/3. So the ratio of the opposite side to the adjacent side is 4:3.

Either by Pythagorean theorem or by past memorization, the hypotenuse will have the value of 5, and we have our 3-4-5 triangle. As we said, Aisha represents the hypotenuse, which has a value of 20N.

Divide that by the calculated value for the hypotenuse to obtain the factor to multiply the other sides.

That factor is 4.

16 N for Lorenzo, 12 N for Saul