Correct Answer: the magnitude
when a vector is multiplied by a scalar, the resulting transformation affects the vector's magnitude and possibly its direction, depending on the scalar value. this operation is fundamental in vector algebra and has practical implications in multiple fields, such as physics, engineering, and computer graphics.
let's consider a vector \(\vec{a}\) in a cartesian coordinate system defined by its components along the basis vectors \(e_1 = (1,0,0)\), \(e_2 = (0,1,0)\), and \(e_3 = (0,0,1)\). the vector can be expressed as \(\vec{a} = a_1 e_1 + a_2 e_2 + a_3 e_3\). when this vector is multiplied by a scalar \(s\), each component of the vector \(a_1, a_2,\) and \(a_3\) is multiplied by \(s\), resulting in a new vector \(s\vec{a} = sa_1 e_1 + sa_2 e_2 + sa_3 e_3\).
the magnitude of the original vector \(\vec{a}\) is calculated as \(\sqrt{a_1^2 + a_2^2 + a_3^2}\). after scalar multiplication, the magnitude of the new vector \(s\vec{a}\) becomes \(\sqrt{(sa_1)^2 + (sa_2)^2 + (sa_3)^2}\), which simplifies to \(|s|\sqrt{a_1^2 + a_2^2 + a_3^2}\). this shows that the magnitude of the vector has been scaled by the absolute value of the scalar \(s\).
the direction of the vector, which is defined by the ratio of its components, is affected differently depending on the scalar \(s\). if \(s\) is positive, the direction of the vector remains unchanged because each component is simply scaled by \(s\). however, if \(s\) is negative, the vector reverses its direction since each component is multiplied by a negative number, effectively pointing the vector in the opposite direction.
in conclusion, multiplying a vector by a scalar changes the vector's magnitude by the absolute value of the scalar and may reverse its direction if the scalar is negative. this operation does not alter the direction if the scalar is positive. understanding these effects is crucial for correctly manipulating vectors in various applications.
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