Why are some of these scalars and others are vectors? What's the difference and how can you tell?

1. a dot (b x c)

2. (a dot b) x (b dot c)

3. (a + b) dot c

4. a x (b dot c)

5. (a x b) dot (b x c)

6. (a + b) x c

7. a dot (b dot c)

8. (a x b) + (b x c)

9. (a x b) - c

10. a x (b x c)

11. (a dot b) + (b dot c)

12. (a dot b) - c

I said that 1, 3, 5, and 11 are scalars. 6, 7, 8 and 9 are vectors. The rest are meaningless.

if a and b are vectors,

ab is a scalar.
a×b is a vector.

the dot product • is only defined for two vectors.
The cross product × is only defined for two vectors.

So,
1,3,5,11 are scalars
6,8,9,10 are vectors
2,4,7,12 are undefined

To determine whether each expression is a scalar or a vector, we need to understand the characteristics of each type.

Scalars:
A scalar is a quantity that has only magnitude and no direction. It represents a single value or number. Scalars can be positive, negative, or zero.

Vectors:
A vector is a quantity that has both magnitude and direction. It represents a physical quantity such as displacement or force. Vectors are typically represented by an arrow, where the length of the arrow indicates the magnitude, and the direction of the arrow represents the direction of the vector.

Now let's analyze each expression and categorize them as either scalars or vectors:

1. a dot (b x c):
This expression is a scalar. The dot product of two vectors yields a scalar value.

2. (a dot b) x (b dot c):
This expression is invalid. The cross product requires vectors to operate on, but both (a dot b) and (b dot c) are scalars.

3. (a + b) dot c:
This expression is a scalar. The dot product of vectors yields a scalar.

4. a x (b dot c):
This expression is invalid. The dot product (b dot c) is a scalar, but the cross product requires vectors to operate on.

5. (a x b) dot (b x c):
This expression is a scalar. The dot product of two vectors yields a scalar.

6. (a + b) x c:
This expression is a vector. The cross product of vectors yields a vector.

7. a dot (b dot c):
This expression is invalid. The dot product (b dot c) is a scalar, and the dot product requires vectors to operate on.

8. (a x b) + (b x c):
This expression is a vector. The sum of two vectors yields a vector.

9. (a x b) - c:
This expression is a vector. The difference of a vector and a vector yields a vector.

10. a x (b x c):
This expression is a vector. The cross product of vectors yields a vector.

11. (a dot b) + (b dot c):
This expression is a scalar. The sum of two scalars yields a scalar.

12. (a dot b) - c:
This expression is invalid. The dot product (a dot b) is a scalar, and the subtraction operation requires vectors to operate on.

So, to summarize:
Scalars: 1, 3, 5, and 11
Vectors: 6, 8, 9, and 10
Invalid: 2, 4, and 12