# Determine (`bar6i`+`bar2j`) · (`bari` - `bar4j` + `bark`)

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You need to multiply each term in sum `6 bari + 2 barj` by the brackets `bar i - 4 bar j + bar k` such that:

`6 bar i(bar i - 4 bar j + bar k) + 2bar j(bar i - 4 bar j + bar k)`

`6bar i*bar i - 24 bar i*bar j + 6 bar i*bar k + 2 bar j*bar i - 8 bar j*bar j + 2 bar j* bar k`

You need to remember that cross products of different unit vectors `bar i*bar j ` , `bar i*bar k, bar j* bar k` yield zeroes since the unit vector are orthogonal, hence:

`6bar i*bar i - 24 bar i*bar j + 6 bar i*bar k + 2 bar j*bar i - 8 bar j*bar j + 2 bar j* bar k = 6(bar i)^2 - 8(bar j)^2`

You need to remember that cross products of like unit vectors `bar i*bar i = 1` and `bar j*bar j = 1` .

`6bar i*bar i - 24 bar i*bar j + 6 bar i*bar k + 2 bar j*bar i - 8 bar j*bar j + 2 bar j* bar k = 6 - 8 = -2`

**Hence, multiplying the vectors `6 bari + 2 barj ` and `bar i - 4 bar j + bar k` yields -2.**