Question

If \( \vec{a} = t\vec{b} \) where \( t<0 \) is a scalar, then:

• A. \( \vec{a}, \vec{b} \) are like vectors and \( |\vec{a}|>|\vec{b}| \)
• B. \( \vec{a}, \vec{b} \) are unlike vectors and \( |\vec{a}|>|\vec{b}| \)
• C. \( \vec{a}, \vec{b} \) are like vectors and \( |\vec{a}|<|\vec{b}| \)
• D. \( \vec{a}, \vec{b} \) are unlike vectors and either \( |\vec{a}| \geq |\vec{b}| \) or \( |\vec{a}|<|\vec{b}| \)

Hint:

When multiplying vectors by a negative scalar, the direction reverses, and the magnitude relationship depends on the value of the scalar.

Answer 1

The Correct Option is D

Step 1: Use the properties of scalar multiplication.
Since \( \vec{a} = t\vec{b} \) and \( t<0 \), we know that \( \vec{a} \) and \( \vec{b} \) are directed in opposite directions (unlike vectors). The magnitude of \( \vec{a} \) is given by: \[ |\vec{a}| = |t| |\vec{b}| = -t|\vec{b}| \quad (\text{since} \ t<0) \] This means that \( |\vec{a}| \geq |\vec{b}| \) depending on the value of \( t \).
Step 2: Analyze the magnitude relationship.
Since \( t \) is negative, \( |\vec{a}| \) could either be greater than or less than \( |\vec{b}| \), depending on the specific value of \( t \).