Work out 𝐢 cross 𝐣.
We see that this is a question about vector products, and we are asked to calculate the vector product between the vectors 𝐢 and 𝐣. Now, 𝐢 is the unit vector in the 𝑥-direction and 𝐣 is the unit vector in the 𝑦-direction. Since we’re asked to calculate a vector product, let’s begin by recalling the general expression for the vector product between two vectors. We’ll call these vectors 𝐀 and 𝐁, and we’ll assume that both lie in the 𝑥𝑦-plane.
We can write the vectors in component form as 𝐀 equals an 𝑥-component, 𝐴 subscript 𝑥 multiplied by 𝐢, plus a 𝑦-component, 𝐴 subscript 𝑦 multiplied by 𝐣, and similarly for vector 𝐁. Then the vector product 𝐀 cross 𝐁 is defined as the 𝑥-component of 𝐀 multiplied by the 𝑦-component of 𝐁 minus the 𝑦-component of 𝐀 multiplied by the 𝑥-component of 𝐁, and this all multiplied by a unit vector 𝐤, which points in the 𝑧-direction. That’s the general case. Now, let’s apply it to our two vectors 𝐢 and 𝐣.
To use this expression, we need to identify the 𝑥- and 𝑦-components of the vectors 𝐢 and 𝐣. We can write the vector 𝐢 as one multiplied by 𝐢 plus zero multiplied by 𝐣. This is saying that 𝐢, the unit vector in the 𝑥-direction, has one unit along 𝑥 and zero units along 𝑦, which obviously makes sense. In the same way, we can rewrite the vector 𝐣 as zero multiplied by 𝐢 plus one multiplied by 𝐣. Then, we need to use our general expression to evaluate the vector product 𝐢 cross 𝐣.
Looking at the first term in the expression, we see that we need the 𝑥-component of the first vector in our cross product, which in our case is 𝐢 multiplied by the 𝑦-component of the second vector in our cross product, which in our case is 𝐣. Now, the 𝑥-component of 𝐢 is one, and the 𝑦-component of 𝐣 is also one. So, we have for our first term one multiplied by one. Then we subtract our second term. Looking again at our expression for the vector product, we see that the second term is given by the 𝑦-component of the first vector 𝐢, which is zero, multiplied by the 𝑥-component of the second vector 𝐣, which is also zero. So, the second term is zero multiplied by zero. This whole thing is then multiplied by the unit vector 𝐤.
The final step is to evaluate this bit here. The first term is one multiplied by one, which gives us one. And the second term is zero multiplied by zero, which gives us zero. So, we have one minus zero, which gives us one. So, we have one multiplied by 𝐤. Of course, we can write this more simply as just 𝐤.
And so, we have arrived at our answer to the question that the result of the vector product 𝐢 cross 𝐣 is 𝐤, the unit vector in the 𝑧-direction.