Question1. $(\vec{i} \times \vec{j}) \cdot \vec{k}=\vec{i} \cdot(\vec{j} \times \vec{k})$ .
2. If $\vec{v}$ and $\vec{w}$ are any two vectors, then $\|\vec{v}+\vec{w}\|=\|\vec{v}\|+\|\vec{w}\|$ .
3. The value of $\vec{v} \cdot(\vec{v} \times \vec{w})$ is always zero.
4. For any scalar $c$ and any vector $\vec{v}$ , we have $\|c \vec{v}\|=c\|\vec{v}\|$ . earn $50 \%$ partial credit for 2 - 3 correct answers. Answers Submit Answers npted this problem 2 times. corded score is $0 \%$ . empts remainina.

Studdy Solution

STEP 1

1. We are given four mathematical statements involving vectors and scalars.
2. We need to determine the truth of each statement.
3. Vector operations such as dot product and cross product follow specific algebraic rules.
4. Norm properties of vectors are based on vector space axioms.

STEP 2

1. Analyze and verify the first statement using vector identities.
2. Analyze and verify the second statement using properties of vector norms.
3. Analyze and verify the third statement using properties of dot and cross products.
4. Analyze and verify the fourth statement using scalar multiplication properties.

STEP 3

Analyze the first statement: $(\vec{i} \times \vec{j}) \cdot \vec{k} = \vec{i} \cdot (\vec{j} \times \vec{k})$ .
- The left-hand side, $(\vec{i} \times \vec{j}) \cdot \vec{k}$ , simplifies to $1$ because $\vec{i} \times \vec{j} = \vec{k}$ and $\vec{k} \cdot \vec{k} = 1$ . - The right-hand side, $\vec{i} \cdot (\vec{j} \times \vec{k})$ , also simplifies to $1$ because $\vec{j} \times \vec{k} = \vec{i}$ and $\vec{i} \cdot \vec{i} = 1$ .
The statement is true.

STEP 4

Analyze the second statement: $\|\vec{v}+\vec{w}\| = \|\vec{v}\| + \|\vec{w}\|$ .
- This statement is generally false due to the triangle inequality, which states $\|\vec{v}+\vec{w}\| \leq \|\vec{v}\| + \|\vec{w}\|$ . - Equality holds only when $\vec{v}$ and $\vec{w}$ are collinear and in the same direction.
The statement is false.

STEP 5

Analyze the third statement: $\vec{v} \cdot (\vec{v} \times \vec{w}) = 0$ .
- The cross product $\vec{v} \times \vec{w}$ is orthogonal to both $\vec{v}$ and $\vec{w}$ . - Therefore, the dot product $\vec{v} \cdot (\vec{v} \times \vec{w})$ is always zero because the dot product of orthogonal vectors is zero.
The statement is true.

STEP 6

Analyze the fourth statement: $\|c \vec{v}\| = c \|\vec{v}\|$ .
- The norm of a scalar multiple of a vector is given by $\|c \vec{v}\| = |c| \|\vec{v}\|$ . - The statement is true only if $c$ is non-negative.
The statement is false for negative $c$ .
The correct answers are:
1. True
2. False
3. True
4. False

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