Question Video: Finding the Unknown Component of a Vector given the Result of Its Cross Product by Another Vector | Nagwa Question Video: Finding the Unknown Component of a Vector given the Result of Its Cross Product by Another Vector | Nagwa

# Question Video: Finding the Unknown Component of a Vector given the Result of Its Cross Product by Another Vector Mathematics

If π = 3π’ β 5π£, π = ππ’ + 5π£, and π Γ π = 50π€, find the value of π.

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### Video Transcript

If the vector π is equal to three π’ minus five π£, π is equal to ππ’ plus five π£, and the cross product of π and π is equal to 50π€, find the value of π.

To answer this question, weβre going to need to recall what we mean by the cross product of two vectors. Essentially, itβs a way of multiplying two vectors. Unlike the dot product, where we get a scalar quantity, when we find the cross product of two vectors, we get a vector. And we say that if π is the three-dimensional vector given by π one, π two, π three and π is the vector π one, π two, π three. Then the cross product of π and π is equal to the vector π two π three minus π three π two, π three π one minus π one π three, and π one π two minus π two π one.

Now, this isnβt a particularly easy formula to remember. And if youβre struggling, you might want to recall that itβs really just the determinant of a three-by-three matrix. So now, we have a formula. Letβs define π one, π two, π three and π one, π two, π three. π one is the horizontal component for π. Itβs three. π two is the vertical component. Thatβs negative five. And thereβs no π€-component. So π three is equal to zero. π one is equal to π. π two is equal to five. And once again, π three is equal to zero.

The first element of the cross product of these two vectors then is π two π three. So thatβs negative five times zero minus π three π two, which is zero multiplied by five. The second element is π three π one, which is zero times π, minus π one π three, which is three times zero. And the third element is π one π two, thatβs three times five, minus π two π one, thatβs negative five multiplied by π. The first and second elements simplify to zero. And the third element of the cross product of our two vectors becomes 15 plus five π. And thatβs because we have minus negative five. So we get plus five.

Now, if we go back to the question, we see that the cross product of π and π is given to us. Weβre told itβs 50π€. Well, another way to represent that is using these sort of triangular brackets. The vector π crossed with π is zero, zero, 50. And we see then that, for the vectors to be equal, 50 must be equal to 15 plus five π. Weβll solve by subtracting 15 from both sides. And that gives us 35 equals five π. We then divide both sides of this equation by five. And we get seven equals π.

And so if π is equal to three π’ minus five π£, π is equal to ππ’ plus five π£, and the cross product of π and π is 50π€, π must be equal to seven.