$1.$ A data analysis team works with a dataset comprising three features: Price, quality, and popularity of various products. Each product can be represented as a vector in a three$-$dimensional vector space, where each dimen sion corresponds to one of the three features. $1.$ Given the following three product vectors: o Product $1$: P$1=(20,8,5)$ o Product $2$: P$2=(15,7,4)$ o Product $3$: P$3=(30,10,7)$ a$.$ Determine if these vectors are linearly independent. $2.$ The team wants to analyze a new product represented by the vector P$4=(25,9,6).$ b$.$ If possible, Express P$4$ as a linear combination of the other three product vectors. $3.$ If P$4$ can be expressed as a linear combination, discuss what this implies about the new product in relation to the original products. Follow$-$up: How could this analysis inform the team$$s strategy for marketing or pricing the new product?