# Let vector x =< 0,2,-3> and vector y= <0,2,-3>  Find the vector u= 7(vector x), vector v=(vector x + vector y)and vector w= 7(vector x) + vector y.

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the vector `bar u` if `bar u = 7 bar x` , such that:

`bar x = 0*bar i + 2*bar j - 3*bar k`

Multiplying by 7 both sides yields:

`7*bar x = 7*0*bar i + 7*2*bar j - 7*3*bar k`

Substituting `bar u` for `7*bar x` yields:

`bar u = 14 bar j - 21 bar k`

You need to find the vector `bar v` if `bar v = bar x + bar y` , such that:

`bar x + bar y = 0*bar i + 2*bar j - 3*bar k + 0*bar i + 2*bar j - 3*bar k`

You need to add like terms, such that:

`bar x + bar y = (0+0)*bar i + (2+2)*bar j - (3+3)*bar k`

`bar x + bar y = 4 bar j - 6 bar k`

Substituting ` bar v` for `bar x + bar y` yields:

`bar v = 4 bar j - 6 bar k`

You need to find the vector `bar w` if `bar w = 7bar x + bar y` , such that:

`bar w = 14 bar j - 21 bar k + 2*bar j - 3*bar k `

`bar w = (14+2)*bar j - (21+3)*bar k`

`bar w = 16 bar j - 24 bar k`

Hence, evaluating the vectors` bar u, bar v` and `bar w` yields `bar u = 14 bar j - 21 bar k` ; `bar v = 4 bar j - 6 bar k` and `bar w = 16 bar j - 24 bar k` .