QuestionAre the following statements true or false?
False
1. For any scalar and any vector , we have .
False
2. If and are any two vectors, then .
False
3. .
True
4. The value of is always zero.
Studdy Solution
STEP 1
What is this asking?
We need to figure out if some statements about vectors and scalars are true or false, specifically about how their magnitudes and products work.
Watch out!
Vector magnitudes and scalar multiplication can be tricky!
Don't forget that the magnitude of a scalar can be negative, so the magnitude of a scaled vector is the absolute value of the scalar times the magnitude of the vector.
Also, remember that vector addition doesn't work like regular addition!
STEP 2
1. Scalar Multiplication and Magnitude
2. Triangle Inequality
3. Scalar Triple Product
4. Dot and Cross Product Properties
STEP 3
Let's **test** the first statement with an example.
Let and .
Then .
STEP 4
Now, let's **calculate** the magnitudes.
We have .
STEP 5
On the other hand, .
Since , the statement is **false**!
The correct formula is .
STEP 6
The second statement looks like the triangle inequality, but it's missing absolute value signs around the scalar!
This statement is generally **false**.
Imagine and pointing in opposite directions.
Their sum will have a smaller magnitude than the sum of their individual magnitudes.
STEP 7
Let's **compute** both sides of the third statement.
Remember , , and .
STEP 8
First, .
So, .
STEP 9
Next, .
So, .
STEP 10
Since both sides equal **1**, the statement is **true**!
STEP 11
The expression results in a vector that is orthogonal (perpendicular) to both and .
STEP 12
The dot product of two orthogonal vectors is always **zero**.
Therefore, is always zero, making the statement **true**!
STEP 13
1. False
2. False
3. True
4. True
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