Question

e. \\(u \times (w \cdot v)\\)

f. \\(w \cdot (v \times v)\\)

g. \\(\frac{1}{v \times w}\\)

h. \\(\frac{1}{\\|v \times v\\|}\\)

Explanation:

🆕 New Concept Discovered: Vector Operations & Meaning
Checking if vector expressions are mathematically valid.

Step 1: Analyze expression (e)

The expression is:

$$ \boldsymbol{u} \times (\boldsymbol{w} \cdot \boldsymbol{v}) $$

1. The dot product \(\boldsymbol{w} \cdot \boldsymbol{v}\) results in a scalar (a real number).
2. This leaves us with \(\boldsymbol{u} \times k\), where \(k\) is a scalar.
3. The cross product \(\times\) is only defined between two vectors, not between a vector and a scalar.

Therefore, expression (e) is meaningless.

Step 2: Analyze expression (f)

The expression is:

$$ \boldsymbol{w} \cdot (\boldsymbol{v} \times \boldsymbol{v}) $$

1. The cross product of any vector with itself is the zero vector:

$$ \boldsymbol{v} \times \boldsymbol{v} = \boldsymbol{0} $$

1. This leaves us with the dot product of two vectors:

$$ \boldsymbol{w} \cdot \boldsymbol{0} $$

1. The dot product of any vector with the zero vector is defined and equals the scalar \(0\).

Therefore, expression (f) is meaningful (its value is \(0\)).

Step 3: Analyze expression (g)

The expression is:

$$ \frac{1}{\boldsymbol{v} \times \boldsymbol{w}} $$

1. The cross product \(\boldsymbol{v} \times \boldsymbol{w}\) results in a vector.
2. Division by a vector is not a defined mathematical operation (we cannot have a vector in the denominator).

Therefore, expression (g) is meaningless.

Step 4: Analyze expression (h)

The expression is:

$$ \frac{1}{\|\boldsymbol{v} \times \boldsymbol{v}\|} $$

1. The cross product \(\boldsymbol{v} \times \boldsymbol{v}\) is the zero vector \(\boldsymbol{0}\).
2. The magnitude of the zero vector is \(0\):

$$ \|\boldsymbol{0}\| = 0 $$

1. This results in division by zero:

$$ \frac{1}{0} $$

Therefore, expression (h) is meaningless due to division by zero.

Answer:

• e. Meaningless (cannot cross a vector with a scalar)
• f. Meaningful (equals \(0\))
• g. Meaningless (cannot divide by a vector)
• h. Meaningless (division by zero)