Question Video: Adding Two Vectors Then Finding the Cross Product of Their Sum by a Third Vector Mathematics

Given that 𝐀 = 7𝐒 + 2𝐣, 𝐁 = βˆ’π’ + 2𝐣, and 𝐂 = 6𝐒 + 6𝐣, determine (𝐂 + 𝐀) Γ— 𝐁.

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Video Transcript

Given that vector 𝐀 is equal to seven 𝐒 plus two 𝐣, vector 𝐁 is equal to negative 𝐒 plus two 𝐣, and vector 𝐂 is equal to six 𝐒 plus six 𝐣, determine the cross product of 𝐂 plus 𝐀 and 𝐁.

In this question, we’re given three vectors in two dimensions in terms of their 𝐒- and 𝐣-components. Firstly, we need to find the sum of vector 𝐂 and vector 𝐀. We then need to find the cross product of this vector and vector 𝐁. Let’s begin by finding the sum of vectors 𝐂 and 𝐀. This is equal to six 𝐒 plus six 𝐣 plus seven 𝐒 plus two 𝐣. We can find the sum of the 𝐒- and 𝐣-components separately. Six 𝐒 plus seven 𝐒 is equal to 13𝐒 and six 𝐣 plus two 𝐣 is eight 𝐣, giving us the vector 13𝐒 plus eight 𝐣.

Next, we need to find the cross product of this vector and vector 𝐁. This is the cross product of 13𝐒 plus eight 𝐣 and negative 𝐒 plus two 𝐣. We recall that for two vectors 𝐌 and 𝐍 in the coordinate plane with 𝐒 and 𝐣 as unit vectors such that 𝐌 is equal to 𝑒𝐒 plus 𝑓𝐣 and 𝐍 is equal to 𝑔𝐒 plus β„Žπ£, then the cross product of vectors 𝐌 and 𝐍 is the determinant of the two-by-two matrix 𝑒, 𝑓, 𝑔, β„Ž multiplied by the unit vector 𝐀. This is equal to π‘’β„Ž minus 𝑔𝑓 multiplied by the unit vector 𝐀.

In this question, we need to find the determinant of the two-by-two matrix 13, eight, negative one, two. This is equal to 13 multiplied by two minus negative one multiplied by eight, giving us 26 plus eight. And multiplying by the unit vector 𝐀, this is equal to 34𝐀. If 𝐀 is equal to seven 𝐒 plus two 𝐣, 𝐁 is equal to negative 𝐒 plus two 𝐣, and 𝐂 is equal to six 𝐒 plus six 𝐣, then the cross product of 𝐂 plus 𝐀 and 𝐁 is 34𝐀.

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