Chapter 1 Section 2: Introduction

Explanation
Yes, it is possible for two quantities to have the same dimensions but different units. This occurs when different units are used to measure the same physical quantity. For example, length can be measured in meters or feet, both of which have the same dimension of distance. However, it is not possible for two quantities to have the same units but different dimensions. The units used to measure a quantity are directly related to its dimensions, so if the units are the same, the dimensions must also be the same.

Explanation
The statement is false because adding two numbers that have the same dimensions, such as two numbers with the dimensions of length [L], does not always result in a meaningful or valid calculation. Adding two lengths together can only be done if they are measured along the same line or axis. For example, adding 6 meters and 3 meters is valid because they are both measured along the same line. However, adding 6 meters and 3 seconds (which also has the dimension of length [L]) does not make sense and cannot be done. Therefore, the statement is false.

Explanation
The units on the left side of the equals sign are consistent with the units on the right side in the equation v = x/t. The units on the left side are meters per second (m/s), which matches the units on the right side where x is in meters (m) and t is in seconds (s).

Explanation
Dimensional analysis cannot be used to determine the integer value of the exponent n. Dimensional analysis is a method used to check the consistency of equations and to derive relationships between variables based on their dimensions. However, it cannot determine the specific numerical values of the variables or exponents in an equation. In this case, since the dimensions of a and t are known and c has no dimensions, dimensional analysis cannot provide any information about the value of n.

Explanation
The statement "I walked 2 miles due north along the beach" involves a vector because it includes both a magnitude (2 miles) and a direction (due north). A vector represents a quantity that has both magnitude and direction, which is the case in this statement. The other statements do not involve vectors as they either only provide a magnitude (2 miles) or do not provide any information about magnitude or direction.

Explanation
The maximum possible value for the magnitude of the resultant vector can be achieved when the two vectors are aligned in the same direction. In this case, the magnitudes are added together, resulting in a magnitude of 3 m + 8 m = 11 m.

The minimum possible value for the magnitude of the resultant vector can be achieved when the two vectors are aligned in opposite directions. In this case, the magnitudes are subtracted, resulting in a magnitude of 8 m - 3 m = 5 m.

Explanation
Two nonzero perpendicular vectors cannot be added together to get a sum of zero. When two vectors are perpendicular, their dot product is always zero. If the sum of two nonzero vectors is zero, it means their components cancel each other out completely. However, since the vectors are perpendicular, their components are independent of each other and cannot cancel out to give a sum of zero. Therefore, the answer is no.

Explanation
Yes, three or more vectors with unequal magnitudes can be added together so their sum is zero. This is possible if the vectors have the right direction and cancel each other out when added. For example, if we have three vectors pointing in different directions but with magnitudes that are inversely proportional to their direction, they can be added together to result in a zero sum.